γλώσσες
Σελίδες
Νομικός
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Numerical methods for stiff Ordinarydifferential equations. Application to the
Finite Element Method (FEM)
Elisabete Alberdi Celaya
EUIT de Minas y Obras Publicas UPV/EHU, Paseo Rafael Moreno Pitxitxi 2,48013 Bilbao (Vizcaya)
April 7, 2013
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Index
1 Introduction
2 Numerical methods for first order ODEs
3 Changing the predictor in EBDF and MEBDF methods
4 Linear multistep methods for second order ODEs
5 BDF-α method
6 Object Oriented Programming methodology
7 Results
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
0
1
Ni(x
j)=δ
ij
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
0
1
Ni(x
j)=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) =∑ n−1
j=2 dj (t)Nj (x)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
0
1
Ni(x
j)=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) =∑ n−1
j=2 dj (t)Nj (x)
Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
0
1
Ni(x
j)=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) =∑ n−1
j=2 dj (t)Nj (x)
Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1
Weak formulation: ∫ L0 Ni ρcput dx =
∫ L0 N′
i kuhx dx, i = 2, ..., n − 1
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
0
1
Ni(x
j)=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) =∑ n−1
j=2 dj (t)Nj (x)
Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1
Weak formulation: ∫ L0 Ni ρcput dx =
∫ L0 N′
i kuhx dx, i = 2, ..., n − 1
Ordinary Differential Equations System:
∑ n−1j=2
∫ L
0ρcpNi (x)Nj (x)dx
︸ ︷︷ ︸
mij
d ′
j (t) = −∑ n−1
j=2
∫ L
0kN′
i (x)N′
j (x)dx
︸ ︷︷ ︸
kij
dj (t), i, j = 2, ..., n − 1
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
0
1
Ni(x
j)=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) =∑ n−1
j=2 dj (t)Nj (x)
Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1
Weak formulation: ∫ L0 Ni ρcput dx =
∫ L0 N′
i kuhx dx, i = 2, ..., n − 1
Ordinary Differential Equations System:
∑ n−1j=2
∫ L
0ρcpNi (x)Nj (x)dx
︸ ︷︷ ︸
mij
d ′
j (t) = −∑ n−1
j=2
∫ L
0kN′
i (x)N′
j (x)dx
︸ ︷︷ ︸
kij
dj (t), i, j = 2, ..., n − 1
DIFFUSION EQUATION:
Md′(t) = α2K d(t),
IC : d0i = g(x i), ∀i ∈ ηd
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A FEM application to the 1D linear diffusionequation
PDEs→ FEM approximation
Difussion:
ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)
BC : u(0, t) = 0 = u(L, t)
IC : u(x , 0) = g(x), ∀x ∈ [0, L]
0
1
Ni(x
j)=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) =∑ n−1
j=2 dj (t)Nj (x)
Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1
Weak formulation: ∫ L0 Ni ρcput dx =
∫ L0 N′
i kuhx dx, i = 2, ..., n − 1
Ordinary Differential Equations System:
∑ n−1j=2
∫ L
0ρcpNi (x)Nj (x)dx
︸ ︷︷ ︸
mij
d ′
j (t) = −∑ n−1
j=2
∫ L
0kN′
i (x)N′
j (x)dx
︸ ︷︷ ︸
kij
dj (t), i, j = 2, ..., n − 1
DIFFUSION EQUATION:
Md′(t) = α2K d(t),
IC : d0i = g(x i), ∀i ∈ ηd
WAVE EQUATION:
Md′′(t) = α2K d(t),
IC : d0i = g1(x i), ∀i ∈ ηd ,
(d ′
i )0
= g2(x i), ∀i ∈ ηd
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
The 1D linear wave equation: L = 8cm, T = 16s, α2 = 1 and three Initial Conditions (g(x)):
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
The 1D linear wave equation: L = 8cm, T = 16s, α2 = 1 and three Initial Conditions (g(x)):
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
Continuous solution: Separation of variables:
ρutt = Tuxx ⇒ u(x, t) =∑
∞
k=1 Ak sin(
kπx8
)cos(ωk t), where:
ωk = kπ
8 , φk = sin(
kπx8
)
Ak = 2L
∫ L0 g(x) sin
(kπx
8
)dx
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
The 1D linear wave equation: L = 8cm, T = 16s, α2 = 1 and three Initial Conditions (g(x)):
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
Continuous solution: Separation of variables:
ρutt = Tuxx ⇒ u(x, t) =∑
∞
k=1 Ak sin(
kπx8
)cos(ωk t), where:
ωk = kπ
8 , φk = sin(
kπx8
)
Ak = 2L
∫ L0 g(x) sin
(kπx
8
)dx
Solution of the discrete model: Modal superposition.
Md′′(t) = −K d(t) ⇒ u(x, t) ≈ uh(x, t) =∑ n−2
k=1 Yk (0)φk (x) cos(ωk t), where:
ωk , φk
Yk (0) =φT
kMgh(x)
φTk
Mφk
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
The 1D linear wave equation: L = 8cm, T = 16s, α2 = 1 and three Initial Conditions (g(x)):
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
Continuous solution: Separation of variables:
ρutt = Tuxx ⇒ u(x, t) =∑
∞
k=1 Ak sin(
kπx8
)cos(ωk t), where:
ωk = kπ
8 , φk = sin(
kπx8
)
Ak = 2L
∫ L0 g(x) sin
(kπx
8
)dx
Solution of the discrete model: Modal superposition.
Md′′(t) = −K d(t) ⇒ u(x, t) ≈ uh(x, t) =∑ n−2
k=1 Yk (0)φk (x) cos(ωk t), where:
ωk , φk
Yk (0) =φT
kMgh(x)
φTk
Mφk
100 element discretization:
0 20 40 60 80 1000
5
10
15
20
25
30
35
40
45
number of the frequence
valu
e of
the
freq
uenc
e
discretcontinuous
Figure: Frequencies of thediscrete and continuous models.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
The 1D linear wave equation: L = 8cm, T = 16s, α2 = 1 and three Initial Conditions (g(x)):
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
Continuous solution: Separation of variables:
ρutt = Tuxx ⇒ u(x, t) =∑
∞
k=1 Ak sin(
kπx8
)cos(ωk t), where:
ωk = kπ
8 , φk = sin(
kπx8
)
Ak = 2L
∫ L0 g(x) sin
(kπx
8
)dx
Solution of the discrete model: Modal superposition.
Md′′(t) = −K d(t) ⇒ u(x, t) ≈ uh(x, t) =∑ n−2
k=1 Yk (0)φk (x) cos(ωk t), where:
ωk , φk
Yk (0) =φT
kMgh(x)
φTk
Mφk
100 element discretization:
0 20 40 60 80 1000
5
10
15
20
25
30
35
40
45
number of the frequence
valu
e of
the
freq
uenc
e
discretcontinuous
Figure: Frequencies of thediscrete and continuous models.
0 1 2 3 4 5 6 7 8−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure: Modes 1, 2 and 10 (continuous and discrete).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
0 1 2 3 4 5 6 7 8−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure: Mode 99 of the continuous.
0 1 2 3 4 5 6 7 8−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure: Mode 99 of the discrete model.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
0 1 2 3 4 5 6 7 8−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure: Mode 99 of the continuous.
0 1 2 3 4 5 6 7 8−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure: Mode 99 of the discrete model.
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
discretecontinuous
Figure: Modal participation factors|Ak |, |Yi (0)| for pulse IC.
52 54 56 58 60 620
0.01
0.02
0.03
0.04
0.05
0.06
discretecontinuous
Figure: Modal participation factors|Ak |, |Yi (0)| for pulse IC (detail).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5
25 modos continuos
desplamiento nodos − tiempo
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5
25 modos continuos
desplamiento nodos − tiempo
DISCRETES
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5
25 modos continuos
desplamiento nodos − tiempo
DISCRETES
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=1599
desplamiento nodos − tiempo
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5
25 modos continuos
desplamiento nodos − tiempo
DISCRETES
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=1599
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=399
desplamiento nodos − tiempo
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5
25 modos continuos
desplamiento nodos − tiempo
DISCRETES
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=1599
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=399
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=99
desplamiento nodos − tiempo
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5
25 modos continuos
desplamiento nodos − tiempo
DISCRETES
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=1599
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=399
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=99
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=25
desplamiento nodos − tiempo
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
CONTINUOUS
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.51599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5399 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.599 modos continuos
desplamiento nodos − tiempo
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5
25 modos continuos
desplamiento nodos − tiempo
DISCRETES
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=1599
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=399
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=99
desplamiento nodos − tiempo
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=1600, nº modos=25
desplamiento nodos − tiempo
t= 0t= 2
The discrete model presents noise because of the high modes. By eliminating high modes,the noise disappears but the solution loses precision.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Integration of the ODE system which comes from FEM.
- Stiffness, makes the solution expensive (mores steps).- Stiffness → existence of eigenvalues of different magnitude in the solution.- Increase of the number of elements, increases stiffness.- Matlab odesuite: ode45, ode15s. Adaptative step size.
Difussion equation:
Md′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
Eigenvalues:100 elements: λmax = −1875, λmin = −0.15421000 elements: λmax = −187500, λmin = −0.1542
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Integration of the ODE system which comes from FEM.
- Stiffness, makes the solution expensive (mores steps).- Stiffness → existence of eigenvalues of different magnitude in the solution.- Increase of the number of elements, increases stiffness.- Matlab odesuite: ode45, ode15s. Adaptative step size.
Difussion equation:
Md′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
Eigenvalues:100 elements: λmax = −1875, λmin = −0.15421000 elements: λmax = −187500, λmin = −0.1542
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 tiempo=16, nele=1000
desplazamiento nodos − tiempo
t= 0t= 2t= 4t= 8t= 16
Senoidal: The ode15s is 83times quicker (lesscomputation time).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Integration of the ODE system which comes from FEM.
- Stiffness, makes the solution expensive (mores steps).- Stiffness → existence of eigenvalues of different magnitude in the solution.- Increase of the number of elements, increases stiffness.- Matlab odesuite: ode45, ode15s. Adaptative step size.
Difussion equation:
Md′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
Eigenvalues:100 elements: λmax = −1875, λmin = −0.15421000 elements: λmax = −187500, λmin = −0.1542
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 tiempo=16, nele=1000
desplazamiento nodos − tiempo
t= 0t= 2t= 4t= 8t= 16
Senoidal: The ode15s is 83times quicker (lesscomputation time).
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1tiempo=16, nele=100
desplamiento nodos − tiempo
t= 0t= 2t= 4t= 8t= 16
Triangular: The ode15s is 114times quicker.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Integration of the ODE system which comes from FEM.
- Stiffness, makes the solution expensive (mores steps).- Stiffness → existence of eigenvalues of different magnitude in the solution.- Increase of the number of elements, increases stiffness.- Matlab odesuite: ode45, ode15s. Adaptative step size.
Difussion equation:
Md′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
Eigenvalues:100 elements: λmax = −1875, λmin = −0.15421000 elements: λmax = −187500, λmin = −0.1542
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 tiempo=16, nele=1000
desplazamiento nodos − tiempo
t= 0t= 2t= 4t= 8t= 16
Senoidal: The ode15s is 83times quicker (lesscomputation time).
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1tiempo=16, nele=100
desplamiento nodos − tiempo
t= 0t= 2t= 4t= 8t= 16
Triangular: The ode15s is 114times quicker.
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1tiempo=16, nele=100
desplamiento nodos − tiempo
t= 0t= 2t= 4t= 8t= 16
Rectangular pulse: Theode15s is 38 times quicker.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Wave equation:
Md′′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
d′(0) = d′0 = (0, ..., 0))T
Eigenvalues:100 elements: λmax = ±43.29i, λmin = ±0.3927i1000 elements: λmax = ±433.01i, λmin = ±0.3927i
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Wave equation:
Md′′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
d′(0) = d′0 = (0, ..., 0))T
Eigenvalues:100 elements: λmax = ±43.29i, λmin = ±0.3927i1000 elements: λmax = ±433.01i, λmin = ±0.3927i
Senoidal:
0 1 2 3 4 5 6 7 8−1.5
−1
−0.5
0
0.5
1
1.5tiempo=16, nele=100
desplazamiento nodos− tiempo
t= 0t= 2t= 4t= 8t= 16
The ode15s is 11 times quicker (it was 83in diffusion).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Wave equation:
Md′′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
d′(0) = d′0 = (0, ..., 0))T
Eigenvalues:100 elements: λmax = ±43.29i, λmin = ±0.3927i1000 elements: λmax = ±433.01i, λmin = ±0.3927i
Senoidal:
0 1 2 3 4 5 6 7 8−1.5
−1
−0.5
0
0.5
1
1.5tiempo=16, nele=100
desplazamiento nodos− tiempo
t= 0t= 2t= 4t= 8t= 16
The ode15s is 11 times quicker (it was 83in diffusion).
Triangular:
0 1 2 3 4 5 6 7 8−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1tiempo=16, nele=100
desplazamiento nodos− tiempo
t= 0t= 2t= 4t= 8t= 10t= 16
The advantage of the ode15s disappears.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
Wave equation:
Md′′(t) = −K d(t),IC : d(0) = d0 = (g(x2), ..., g(xn−1))
T
d′(0) = d′0 = (0, ..., 0))T
Eigenvalues:100 elements: λmax = ±43.29i, λmin = ±0.3927i1000 elements: λmax = ±433.01i, λmin = ±0.3927i
Senoidal:
0 1 2 3 4 5 6 7 8−1.5
−1
−0.5
0
0.5
1
1.5tiempo=16, nele=100
desplazamiento nodos− tiempo
t= 0t= 2t= 4t= 8t= 16
The ode15s is 11 times quicker (it was 83in diffusion).
Triangular:
0 1 2 3 4 5 6 7 8−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1tiempo=16, nele=100
desplazamiento nodos− tiempo
t= 0t= 2t= 4t= 8t= 10t= 16
The advantage of the ode15s disappears.
Pulse: The advantage of theode15s disappears.
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método ode15s, nele=400, pasos=12837, masa=cons
desplazamiento nodos − tiempo
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
400 elements:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
400 elements:
Ode15s, 12837 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método ode15s, nele=400, pasos=12837, masa=cons
desplazamiento nodos − tiempo
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
400 elements:
Ode15s, 12837 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método ode15s, nele=400, pasos=12837, masa=cons
desplazamiento nodos − tiempo
t= 0t= 2
Modal superposition:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=400, nº modos= 399
desplamiento nodos − tiempo
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
400 elements:
Ode15s, 12837 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método ode15s, nele=400, pasos=12837, masa=cons
desplazamiento nodos − tiempo
t= 0t= 2
Modal superposition:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=400, nº modos= 399
desplamiento nodos − tiempo
t= 0t= 2
HHT-α method (“α” method),1400 steps:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
400 elements:
Ode15s, 12837 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método ode15s, nele=400, pasos=12837, masa=cons
desplazamiento nodos − tiempo
t= 0t= 2
Modal superposition:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=400, nº modos= 399
desplamiento nodos − tiempo
t= 0t= 2
HHT-α method (“α” method),1400 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método HHT−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=0.3 γ=0.8 β=0.4225
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
400 elements:
Ode15s, 12837 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método ode15s, nele=400, pasos=12837, masa=cons
desplazamiento nodos − tiempo
t= 0t= 2
Modal superposition:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=400, nº modos= 399
desplamiento nodos − tiempo
t= 0t= 2
HHT-α method (“α” method),1400 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método HHT−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=0.3 γ=0.8 β=0.4225
t= 0t= 2
Newmark’s method β = 1/6,γ = 0.5, 800 steps →Superconvergence:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear diffusion and wave equation examples inMATLAB
400 elements:
Ode15s, 12837 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método ode15s, nele=400, pasos=12837, masa=cons
desplazamiento nodos − tiempo
t= 0t= 2
Modal superposition:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método supmod., nele=400, nº modos= 399
desplamiento nodos − tiempo
t= 0t= 2
HHT-α method (“α” method),1400 steps:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método HHT−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=0.3 γ=0.8 β=0.4225
t= 0t= 2
Newmark’s method β = 1/6,γ = 0.5, 800 steps →Superconvergence:
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método Newmark, tiempo=16, nele=400, pasos=800, masa=cons
γ=0.5, β=1/6
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A non-linear version of the wave equation
Non linear PDE of a guitar string:
ρutt (x, t) =
T + E · S(√
1 + u2x (x, t) − 1
)
︸ ︷︷ ︸
T
uxx (x, t)
Real data: L = 0.648 m, diam = 0.41 · 10−3 m,S = 0.25 · π · diam2, frec = 329,T = 1.8002 · 102.IC: First mode of vibration.Time interval: 1 and 5 linear periods (5 linearperiods=0.015198 s)
20 elements are considered:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A non-linear version of the wave equation
Non linear PDE of a guitar string:
ρutt (x, t) =
T + E · S(√
1 + u2x (x, t) − 1
)
︸ ︷︷ ︸
T
uxx (x, t)
Real data: L = 0.648 m, diam = 0.41 · 10−3 m,S = 0.25 · π · diam2, frec = 329,T = 1.8002 · 102.IC: First mode of vibration.Time interval: 1 and 5 linear periods (5 linearperiods=0.015198 s)
20 elements are considered:
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método trap., tiempo=0.0030395, nele=20, pasos=200
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método ode15s, tiempo=0.0030395, nele=20, pasos=1410
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A non-linear version of the wave equation
Non linear PDE of a guitar string:
ρutt (x, t) =
T + E · S(√
1 + u2x (x, t) − 1
)
︸ ︷︷ ︸
T
uxx (x, t)
Real data: L = 0.648 m, diam = 0.41 · 10−3 m,S = 0.25 · π · diam2, frec = 329,T = 1.8002 · 102.IC: First mode of vibration.Time interval: 1 and 5 linear periods (5 linearperiods=0.015198 s)
20 elements are considered:
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método trap., tiempo=0.0030395, nele=20, pasos=200
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método ode15s, tiempo=0.0030395, nele=20, pasos=1410
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.0030395, nele=20, pasos=200, masa=cons
α=0.3 , γ=0.8, β=0.4225
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A non-linear version of the wave equation
Non linear PDE of a guitar string:
ρutt (x, t) =
T + E · S(√
1 + u2x (x, t) − 1
)
︸ ︷︷ ︸
T
uxx (x, t)
Real data: L = 0.648 m, diam = 0.41 · 10−3 m,S = 0.25 · π · diam2, frec = 329,T = 1.8002 · 102.IC: First mode of vibration.Time interval: 1 and 5 linear periods (5 linearperiods=0.015198 s)
20 elements are considered:
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método trap., tiempo=0.0030395, nele=20, pasos=200
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método ode15s, tiempo=0.0030395, nele=20, pasos=1410
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.0030395, nele=20, pasos=200, masa=cons
α=0.3 , γ=0.8, β=0.4225
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método trap., tiempo=0.015198, nele=20, pasos=1000
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método ode15s, tiempo=0.015198, nele=20, pasos=9425
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
A non-linear version of the wave equation
Non linear PDE of a guitar string:
ρutt (x, t) =
T + E · S(√
1 + u2x (x, t) − 1
)
︸ ︷︷ ︸
T
uxx (x, t)
Real data: L = 0.648 m, diam = 0.41 · 10−3 m,S = 0.25 · π · diam2, frec = 329,T = 1.8002 · 102.IC: First mode of vibration.Time interval: 1 and 5 linear periods (5 linearperiods=0.015198 s)
20 elements are considered:
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método trap., tiempo=0.0030395, nele=20, pasos=200
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método ode15s, tiempo=0.0030395, nele=20, pasos=1410
0 0.5 1 1.5 2 2.5 3 3.5
x 10−3
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.0030395, nele=20, pasos=200, masa=cons
α=0.3 , γ=0.8, β=0.4225
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método trap., tiempo=0.015198, nele=20, pasos=1000
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método ode15s, tiempo=0.015198, nele=20, pasos=9425
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.015198, nele=20, pasos=1000, masa=cons
α=0.3 , γ=0.8, β=0.4225
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Numerical methods for first order ODEs
A first order ODE is given by: y ′(t) = f (t, y(t)), y(a) = y0
Runge-Kutta methods ⇒ ode45
Linear multistep methods ⇒ BDFs ⇒ ode15s
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Numerical methods for first order ODEs
A first order ODE is given by: y ′(t) = f (t, y(t)), y(a) = y0
Runge-Kutta methods ⇒ ode45
Linear multistep methods ⇒ BDFs ⇒ ode15s
Search of better linear multistep methods
The search of linear multistep methods with better stability and precision characteristicsfollowing 3 directions:
using high order derivatives
using superfuture-points
combining two existing methods or techniques to generate them
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Error, stability and stiffness
Amplification factor
A method is stable if the perturbations are not amplified. Apply the method to the testequation: y ′ = λy .- Linear multistep method:
∑ kj=0 αj yn+j = h
∑ kj=0 βj yn+j , where h = λh ⇒
yn+1yn+2...
yn+k
=
a11 a12 . . . a1ka21 a22 . . . a2k
...... · · ·
...ak1 ak2 . . . akk
·
ynyn+1...
yn+k−1
⇒ Yn+k = A
(
h)
Yn+k−1
where: Yn+k = (yn+1, yn+2, ..., yn+k )T , Yn+k−1 = (yn, yn+1, ..., yn+k−1)T and A the
amplification factor.
- One-step method ⇒ Matrix A is an escalar function: yn+1 = R(
h)
yn
Numerical stability: The module of the eigenvalues of A is less than or equal to 1.
The spectral radius is the maximum module of the eigenvalues:ρ = max |ρi | : ρi eigenvalue of A
Stability region:
S =
h ∈ C :∣∣∣rj
(
h)∣∣∣ ≤ 1 ∀ h, rj root of the characteristic polynomial of A
The frontier of the stability region: h = hλ : r(h) = 1. To draw it we do: r = eiθ andθ ∈ [0, 2pi).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Error, stability and stiffness
0
0
Figure: A-stability (or unconditional stability).
0
0α
−α
Figure: A(α) stability.
Precision of a method
Global truncation error: GTEn+k = y(tn+k ) − yn+kLocal truncation error: LTEn+k = y(tn+k ) − y∗
n+kLocalizing assumption to calculate y∗
n+k : yn+j = y(tn+j ), for j = 0, 1, ..., k − 1Method of order p: LTE = O(hp+1) ⇒ GTE = O(hp)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Runge-Kutta methods
yn+1 = yn + hs∑
i=1
biki , where: ki = f (tn + cih, yn + hs∑
j=1aijkj), i = 1, 2, ...., s
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Runge-Kutta methods
yn+1 = yn + hs∑
i=1
biki , where: ki = f (tn + cih, yn + hs∑
j=1aijkj), i = 1, 2, ...., s
b = [b1, b2, ..., bs]T
, c = [c1, c2, ..., cs]T
, A =[aij]
c A
bT
Table: Butcher Table.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Runge-Kutta methods
yn+1 = yn + hs∑
i=1
biki , where: ki = f (tn + cih, yn + hs∑
j=1aijkj), i = 1, 2, ...., s
b = [b1, b2, ..., bs]T
, c = [c1, c2, ..., cs]T
, A =[aij]
c A
bT
Table: Butcher Table.
Stability:
yn+1 = R(h)yn, h = hλ
R(h) = 1 + hbT(
I − hA)−1
e, e = [1, 1, ..., 1]T ∈ Rs
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Runge-Kutta methods
yn+1 = yn + hs∑
i=1
biki , where: ki = f (tn + cih, yn + hs∑
j=1aijkj), i = 1, 2, ...., s
b = [b1, b2, ..., bs]T
, c = [c1, c2, ..., cs]T
, A =[aij]
c A
bT
Table: Butcher Table.
Stability:
yn+1 = R(h)yn, h = hλ
R(h) = 1 + hbT(
I − hA)−1
e, e = [1, 1, ..., 1]T ∈ Rs
−4 −3 −2 −1 0 1−3i
−2i
−i
0
i
2i
3i
p=1
p=2
p=3
p=4
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Runge-Kutta methods
yn+1 = yn + hs∑
i=1
biki , where: ki = f (tn + cih, yn + hs∑
j=1aijkj), i = 1, 2, ...., s
b = [b1, b2, ..., bs]T
, c = [c1, c2, ..., cs]T
, A =[aij]
c A
bT
Table: Butcher Table.
Stability:
yn+1 = R(h)yn, h = hλ
R(h) = 1 + hbT(
I − hA)−1
e, e = [1, 1, ..., 1]T ∈ Rs
−4 −3 −2 −1 0 1−3i
−2i
−i
0
i
2i
3i
p=1
p=2
p=3
p=4
Embedded Runge-Kutta methods: Methods oforder p and p + 1 share the coefficients ci , aij .DOPRI(5,4)→ ode45.
c A
bT
bT
ET
Table: Embedded Runge-Kutta Butcher Table.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Runge-Kutta methods
yn+1 = yn + hs∑
i=1
biki , where: ki = f (tn + cih, yn + hs∑
j=1aijkj), i = 1, 2, ...., s
b = [b1, b2, ..., bs]T
, c = [c1, c2, ..., cs]T
, A =[aij]
c A
bT
Table: Butcher Table.
Stability:
yn+1 = R(h)yn, h = hλ
R(h) = 1 + hbT(
I − hA)−1
e, e = [1, 1, ..., 1]T ∈ Rs
−4 −3 −2 −1 0 1−3i
−2i
−i
0
i
2i
3i
p=1
p=2
p=3
p=4
Embedded Runge-Kutta methods: Methods oforder p and p + 1 share the coefficients ci , aij .DOPRI(5,4)→ ode45.
c A
bT
bT
ET
Table: Embedded Runge-Kutta Butcher Table.
−6 −4 −2 0 2−4i
−3i
−2i
−i
0
i
2i
3i
4i
p=4
p=5
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods
Linear multistep methods:k∑
j=0
αj yn+j = hk∑
j=0
βj fn+j
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods
Linear multistep methods:k∑
j=0
αj yn+j = hk∑
j=0
βj fn+j
-Backward Differentiation Formulae (BDF):∑ k
j=11j ∇
j yn+k = hfn+k
-Numerical Differentiation Formulae (NDF):∑ k
j=11j ∇
j yn+k = hfn+k + κ∇k+1yn+k
−10 −5 0 5 10 15 20−15i
−10i
−5i
0
5i
10i
15i
BDF2BDF3
BDF4
BDF5
BDF1
Figure: BDF stability regions(exterior to the curves).
k κ NDF %step size BDF’s A(α) NDF’s A(α)1 -0.1850 26% 90 902 -1/9 26% 90 903 -0.0823 26% 86 804 -0.0415 12% 73 66
Table: NDFs: efficiency and stability respect to BDFs.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods
Linear multistep methods:k∑
j=0
αj yn+j = hk∑
j=0
βj fn+j
-Backward Differentiation Formulae (BDF):∑ k
j=11j ∇
j yn+k = hfn+k
-Numerical Differentiation Formulae (NDF):∑ k
j=11j ∇
j yn+k = hfn+k + κ∇k+1yn+k
−10 −5 0 5 10 15 20−15i
−10i
−5i
0
5i
10i
15i
BDF2BDF3
BDF4
BDF5
BDF1
Figure: BDF stability regions(exterior to the curves).
k κ NDF %step size BDF’s A(α) NDF’s A(α)1 -0.1850 26% 90 902 -1/9 26% 90 903 -0.0823 26% 86 804 -0.0415 12% 73 66
Table: NDFs: efficiency and stability respect to BDFs.
Some modifications to the linear multistep methods:- Extended BDF (EBDF):
∑ kj=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1
- Modified Extended BDF (MEBDF):∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1 + h(
βk − βk
)
fn+k
Both A-stable up to order 4.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Changing the predictor in EBDF and MEBDFmethods
The motivation of the change is double
- NDF-s imply few computational additional cost with respect to BDFs.- Good stability characteristics of EBDF and MEBDF.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Changing the predictor in EBDF and MEBDFmethods
The motivation of the change is double
- NDF-s imply few computational additional cost with respect to BDFs.- Good stability characteristics of EBDF and MEBDF.
Predictor-corrector scheme of EBDF and MEBDF:
Predict yn+k using the k step BDF.
Predict yn+k+1 of the instant tn+k+1 using the k step BDF.
Evaluate fn+k+1 = f (tn+k+1, yn+k+1) and also fn+k = f (tn+k , yn+k ) for MEBDFs.
Substitute these values in the correctors:EBDF:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1
MEBDF:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1 + h(
βk − βk
)
fn+k
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Changing the predictor in EBDF and MEBDFmethods
The motivation of the change is double
- NDF-s imply few computational additional cost with respect to BDFs.- Good stability characteristics of EBDF and MEBDF.
Predictor-corrector scheme of EBDF and MEBDF:
Predict yn+k using the k step BDF.
Predict yn+k+1 of the instant tn+k+1 using the k step BDF.
Evaluate fn+k+1 = f (tn+k+1, yn+k+1) and also fn+k = f (tn+k , yn+k ) for MEBDFs.
Substitute these values in the correctors:EBDF:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1
MEBDF:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1 + h(
βk − βk
)
fn+k
Lemma
If the predictors used are of order k and the correctors of order k + 1, the whole algorithm isof order (k + 1).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Changing the predictor in EBDF and MEBDFmethods
The motivation of the change is double
- NDF-s imply few computational additional cost with respect to BDFs.- Good stability characteristics of EBDF and MEBDF.
Predictor-corrector scheme of EBDF and MEBDF:
Predict yn+k using the k step BDF.
Predict yn+k+1 of the instant tn+k+1 using the k step BDF.
Evaluate fn+k+1 = f (tn+k+1, yn+k+1) and also fn+k = f (tn+k , yn+k ) for MEBDFs.
Substitute these values in the correctors:EBDF:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1
MEBDF:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1 + h(
βk − βk
)
fn+k
Lemma
If the predictors used are of order k and the correctors of order k + 1, the whole algorithm isof order (k + 1).
Local truncation errors: EBDF: LTEk = hk+2
βk+1C1
(
1 −αk−1
αk
)∂f
∂yy (k+1)
+ C3y (k+2)
(tn) + O(
hk+3)
MEBDF: LTEk = hk+2
C1
(
βk+1
(
1 −αk−1
αk
)
+ (βk − βk )
)∂f
∂yy (k+1)
+ C4y (k+2)
(tn) + O(
hk+3)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Order and stability of EBDF and MEBDF
Stability:The characteristic polynomial in both cases: Ah3 + Bh2 + Ch + D = 0
where:
A = −βk r k
B = 2αk βk r k + T − βk+1S−(βk − βk )RC = −βk α2
k r k − 2αk T + αk βk+1S − βk+1αk−1R+(βk − βk )Rαk
D = α2k T
R =∑ k−1
j=0 αj rj , S =
∑ k−2j=0 αj r
j+1, T =∑ k
j=0 αj rj ,
The red coefficients are substituted by βk in EBDFs.The blue coefficients are characteristic of MEBDFs.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Order and stability of EBDF and MEBDF
Stability:The characteristic polynomial in both cases: Ah3 + Bh2 + Ch + D = 0
where:
A = −βk r k
B = 2αk βk r k + T − βk+1S−(βk − βk )RC = −βk α2
k r k − 2αk T + αk βk+1S − βk+1αk−1R+(βk − βk )Rαk
D = α2k T
R =∑ k−1
j=0 αj rj , S =
∑ k−2j=0 αj r
j+1, T =∑ k
j=0 αj rj ,
The red coefficients are substituted by βk in EBDFs.The blue coefficients are characteristic of MEBDFs.
Scheme of the new methods:
Mantaining the corrector:- Two NDF predictors ⇒ ENDF and MENDF.- First corrector BDF and second corrector NDF ⇒ EBNDF and MEBNDF.- First corrector NDF and second corrector BDF ⇒ ENBDF and MENBDF.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Order and stability of EBDF and MEBDF
Stability:The characteristic polynomial in both cases: Ah3 + Bh2 + Ch + D = 0
where:
A = −βk r k
B = 2αk βk r k + T − βk+1S−(βk − βk )RC = −βk α2
k r k − 2αk T + αk βk+1S − βk+1αk−1R+(βk − βk )Rαk
D = α2k T
R =∑ k−1
j=0 αj rj , S =
∑ k−2j=0 αj r
j+1, T =∑ k
j=0 αj rj ,
The red coefficients are substituted by βk in EBDFs.The blue coefficients are characteristic of MEBDFs.
Scheme of the new methods:
Mantaining the corrector:- Two NDF predictors ⇒ ENDF and MENDF.- First corrector BDF and second corrector NDF ⇒ EBNDF and MEBNDF.- First corrector NDF and second corrector BDF ⇒ ENBDF and MENBDF.
All the methods are of order p = k + 1:
LTEk = hk+2
(
βk+1Ak + Ci (βk − βk )) ∂f
∂yy (k+1)
+ Di y(k+2)
(tn) + O(
hk+3)
where:- the boxed part depends on the predictors.- The coefficients Ak depend on the error coefficients of the predictors.- The blue coefficients are characteristic of MEBDFs.- The coefficient Di depends on the correctors.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Order and stability of EBDF and MEBDF
k p (order) A(α) EBDF A(α) EBNDF A(α) ENBDF A(α) ENDF1 2 90 90 90 902 3 90 90 90 903 4 90 90 90 904 5 87.61 87.68 87.49 87.54
Table: A(α)-estabilidad de los mtodos EBDF, EBNDF,ENBDF, ENDF.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Order and stability of EBDF and MEBDF
k p (order) A(α) EBDF A(α) EBNDF A(α) ENBDF A(α) ENDF1 2 90 90 90 902 3 90 90 90 903 4 90 90 90 904 5 87.61 87.68 87.49 87.54
Table: A(α)-estabilidad de los mtodos EBDF, EBNDF,ENBDF, ENDF.
−1 0 1 2 3 4 5 6 7
−4i
−3i
−2i
−i
0
i
2i
3i
4ik=4
k=3
k=2
k=1
EBDFENDF
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Order and stability of EBDF and MEBDF
k p (order) A(α) EBDF A(α) EBNDF A(α) ENBDF A(α) ENDF1 2 90 90 90 902 3 90 90 90 903 4 90 90 90 904 5 87.61 87.68 87.49 87.54
Table: A(α)-estabilidad de los mtodos EBDF, EBNDF,ENBDF, ENDF.
−1 0 1 2 3 4 5 6 7
−4i
−3i
−2i
−i
0
i
2i
3i
4ik=4
k=3
k=2
k=1
EBDFENDF
k p (order) A(α) MEBDF A(α) MEBNDF A(α) MENBDF A(α) MENDF1 2 90 90 90 902 3 90 90 90 903 4 90 90 90 904 5 88.36 88.41 88.88 88.93
Table: A(α)-estabilidad de los mtodos MEBDF, MEBNDF,MENBDF, MENDF.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Order and stability of EBDF and MEBDF
k p (order) A(α) EBDF A(α) EBNDF A(α) ENBDF A(α) ENDF1 2 90 90 90 902 3 90 90 90 903 4 90 90 90 904 5 87.61 87.68 87.49 87.54
Table: A(α)-estabilidad de los mtodos EBDF, EBNDF,ENBDF, ENDF.
−1 0 1 2 3 4 5 6 7
−4i
−3i
−2i
−i
0
i
2i
3i
4ik=4
k=3
k=2
k=1
EBDFENDF
k p (order) A(α) MEBDF A(α) MEBNDF A(α) MENBDF A(α) MENDF1 2 90 90 90 902 3 90 90 90 903 4 90 90 90 904 5 88.36 88.41 88.88 88.93
Table: A(α)-estabilidad de los mtodos MEBDF, MEBNDF,MENBDF, MENDF.
−1 0 1 2 3 4 5 6 7
−4i
−3i
−2i
−i
0
i
2i
3i
4i
k=1
k=3
k=2
MEBDF
MENDF
k=4
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods for second orderODEs
Stiffness
The second order ODE systems obtained after discretizing the wave-type PDE by the FEMare stiff. The high modes are result of the discretization and they are not representative. Theirresolution requires:- the use of good stability numerical methods.- controlled numerical dissipation in the range of the high frequencies.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods for second orderODEs
Stiffness
The second order ODE systems obtained after discretizing the wave-type PDE by the FEMare stiff. The high modes are result of the discretization and they are not representative. Theirresolution requires:- the use of good stability numerical methods.- controlled numerical dissipation in the range of the high frequencies.
Alfa-generalized method: Different weighting of the inertia forces and the rest of the addends:
Man+1−αm + Cvn+1−αf+ Kdn+1−αf
= F(
tn+1−αf
)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
where:
dn+1−αf= (1 − αf ) dn+1 + αf dn
vn+1−αf= (1 − αf ) vn+1 + αf vn
an+1−αm = (1 − αm) an+1 + αman
tn+1−αf= (1 − αf ) tn+1 + αf tn
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods for second orderODEs
Stiffness
The second order ODE systems obtained after discretizing the wave-type PDE by the FEMare stiff. The high modes are result of the discretization and they are not representative. Theirresolution requires:- the use of good stability numerical methods.- controlled numerical dissipation in the range of the high frequencies.
Alfa-generalized method: Different weighting of the inertia forces and the rest of the addends:
Man+1−αm + Cvn+1−αf+ Kdn+1−αf
= F(
tn+1−αf
)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
where:
dn+1−αf= (1 − αf ) dn+1 + αf dn
vn+1−αf= (1 − αf ) vn+1 + αf vn
an+1−αm = (1 − αm) an+1 + αman
tn+1−αf= (1 − αf ) tn+1 + αf tn
- If αm = 0 ⇒ HHT-α method:
Man+1 + Cvn+1−αf+ Kdn+1−αf
= F(
tn+1−αf
)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods for second orderODEs
Stiffness
The second order ODE systems obtained after discretizing the wave-type PDE by the FEMare stiff. The high modes are result of the discretization and they are not representative. Theirresolution requires:- the use of good stability numerical methods.- controlled numerical dissipation in the range of the high frequencies.
Alfa-generalized method: Different weighting of the inertia forces and the rest of the addends:
Man+1−αm + Cvn+1−αf+ Kdn+1−αf
= F(
tn+1−αf
)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
where:
dn+1−αf= (1 − αf ) dn+1 + αf dn
vn+1−αf= (1 − αf ) vn+1 + αf vn
an+1−αm = (1 − αm) an+1 + αman
tn+1−αf= (1 − αf ) tn+1 + αf tn
- If αm = 0 ⇒ HHT-α method:
Man+1 + Cvn+1−αf+ Kdn+1−αf
= F(
tn+1−αf
)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
- If αf = αm = 0 ⇒ Newmark method:
Man+1 + Cvn+1 + Kdn+1 = F (tn+1)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Linear multistep methods for second orderODEs
Stiffness
The second order ODE systems obtained after discretizing the wave-type PDE by the FEMare stiff. The high modes are result of the discretization and they are not representative. Theirresolution requires:- the use of good stability numerical methods.- controlled numerical dissipation in the range of the high frequencies.
Alfa-generalized method: Different weighting of the inertia forces and the rest of the addends:
Man+1−αm + Cvn+1−αf+ Kdn+1−αf
= F(
tn+1−αf
)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
where:
dn+1−αf= (1 − αf ) dn+1 + αf dn
vn+1−αf= (1 − αf ) vn+1 + αf vn
an+1−αm = (1 − αm) an+1 + αman
tn+1−αf= (1 − αf ) tn+1 + αf tn
- If αm = 0 ⇒ HHT-α method:
Man+1 + Cvn+1−αf+ Kdn+1−αf
= F(
tn+1−αf
)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
- If αf = αm = 0 ⇒ Newmark method:
Man+1 + Cvn+1 + Kdn+1 = F (tn+1)
dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]
vn+1 = vn + ∆t [(1 − γ) an + γan+1]
Order of precision: Second order
- Generalized-alfa: γ = −αm + αf + 12
- HHT-α: γ = αf + 12
- Newmark: γ = 12
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability and spectral radius
Stability study → Applying the method to the second order test equation: u′′ + ω2u = 0
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability and spectral radius
Stability study → Applying the method to the second order test equation: u′′ + ω2u = 0Newmark method: stability and dissipation of high frequencies
- Unconditionally stable: 12 ≤ γ < 2β
- Conditionally stable: γ ≥ 12 and γ > 2β
- Dissipation of high frequencies ρ∞ < 1: β =
(γ+ 1
2
)2
4 . There is not high frequencydissipation in second order Newmark method.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability and spectral radius
Stability study → Applying the method to the second order test equation: u′′ + ω2u = 0Newmark method: stability and dissipation of high frequencies
- Unconditionally stable: 12 ≤ γ < 2β
- Conditionally stable: γ ≥ 12 and γ > 2β
- Dissipation of high frequencies ρ∞ < 1: β =
(γ+ 1
2
)2
4 . There is not high frequencydissipation in second order Newmark method.
HHT-α method: stability and dissipation of high frequencies
Unconditionally stable and dissipation of high frequencies: α ∈[0, 1
3
]and β =
(1+α)2
4
10−2
10−1
100
101
102
103
104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ω/(2π)
ρ
Collocation(γ=0.5,β=0.16,θ=1.514951)
Houbolt
(γ=0.5,β=0.18,θ=1.287301)
(γ=0.5,β=1/6,θ=1.4)Wilson
Collocation
Newmark
TrapezoidalHHT−
(β=0.3025,γ=0.6)
α (α= 0.05)
α (α= 0.3)HHT−
EDMC−1 χ1=χ
2=0.2998
Figure: Spectral radius of some methods as function of Ω/(2π).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
BDF-α method: linear multistep method withcontrolled numerical dissipation
Spectral radius of the BDFs:Second order ODEs ⇒ test equation u′′ + ω2u = 0This second order test equation is transformed in an equivalent first order ODE system:
(uu′
)′
=
(0 1
−ω2 0
) (uu′
)
⇒ y ′= ±iωy
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
BDF-α method: linear multistep method withcontrolled numerical dissipation
Spectral radius of the BDFs:Second order ODEs ⇒ test equation u′′ + ω2u = 0This second order test equation is transformed in an equivalent first order ODE system:
(uu′
)′
=
(0 1
−ω2 0
) (uu′
)
⇒ y ′= ±iωy
Apply the method to the test equation y ′ = λy , where λ = ±iω:
Yn+k = A(
h)
· Yn+k−1
where h = hλ, Yn+k = (yn+1, yn+2, ..., yn+k )T , Yn+k−1 = (yn, yn+1, ..., yn+k−1)T and A the
amplification matrix.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
BDF-α method: linear multistep method withcontrolled numerical dissipation
Spectral radius of the BDFs:Second order ODEs ⇒ test equation u′′ + ω2u = 0This second order test equation is transformed in an equivalent first order ODE system:
(uu′
)′
=
(0 1
−ω2 0
) (uu′
)
⇒ y ′= ±iωy
Apply the method to the test equation y ′ = λy , where λ = ±iω:
Yn+k = A(
h)
· Yn+k−1
where h = hλ, Yn+k = (yn+1, yn+2, ..., yn+k )T , Yn+k−1 = (yn, yn+1, ..., yn+k−1)T and A the
amplification matrix.
The eigenvalues of A and the spectral radiusare calculated → BDF-s have high dissipationof the high frequency modes.
10−2
10−1
100
101
102
103
104
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Ω/(2π)
ρ
(β=0.3025,γ=0.6)
BDF3
BDF5
BDF1
Houbolt
BDF4
HHT−
BDF2(γ=0.5,β=0.16,θ=1.514951)
Park
HHT−α (α= 0.3)
Newmark
α (α= 0.05)
Collocation
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
New method based on the BDF2
BDF2: 32 yn+2 − 2yn+1 + 1
2 yn = hfn+2
- Second order and A-stable.- With a bigger range of spectral radius ρ∞ than the BDF2.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
New method based on the BDF2
BDF2: 32 yn+2 − 2yn+1 + 1
2 yn = hfn+2
- Second order and A-stable.- With a bigger range of spectral radius ρ∞ than the BDF2.
Expression of the method: Weighting with 3 free parameters:32 ((1 + β)yn+2 − βyn+1) − 2 ((1 + γ)yn+1 − γyn) + 1
2 yn = h ((1 + α)fn+2 − αfn+1)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
New method based on the BDF2
BDF2: 32 yn+2 − 2yn+1 + 1
2 yn = hfn+2
- Second order and A-stable.- With a bigger range of spectral radius ρ∞ than the BDF2.
Expression of the method: Weighting with 3 free parameters:32 ((1 + β)yn+2 − βyn+1) − 2 ((1 + γ)yn+1 − γyn) + 1
2 yn = h ((1 + α)fn+2 − αfn+1)
Reagrouping terms it results a linear multistep method:∑ 2
j=0 αj yn+j = h∑ 2
j=0 βj fn+j
where :
α2 = 32 (1 + β), α1 = − 3
2 β − 2(1 + γ), α0 = 2γ + 12
β2 = 1 + α, β1 = −α, β0 = 0
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
Order of precision: LTE = C0y(tn) + C1hy ′(tn) + C2h2y ′′(tn) + ... + Cqhqy (q)(tn + ...
where:
C0 =∑ k
i=0 αi
C1 =∑ k
i=0 iαi −∑ k
i=0 βi
Cq = 1q!
(∑ k
i=0 iqαi
)
− 1(q−1)!
(∑ k
i=0 iq−1βi
)
, q ≥ 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
Order of precision: LTE = C0y(tn) + C1hy ′(tn) + C2h2y ′′(tn) + ... + Cqhqy (q)(tn + ...
where:
C0 =∑ k
i=0 αi
C1 =∑ k
i=0 iαi −∑ k
i=0 βi
Cq = 1q!
(∑ k
i=0 iqαi
)
− 1(q−1)!
(∑ k
i=0 iq−1βi
)
, q ≥ 2
C0 =∑ 2
i=0 αi = 0C1 =
∑ 2i=0 iαi −
∑ 2i=0 βi = −2γ + 3
2 β
C2 = 12!
(∑ 2
i=0 i2αi
)
−(∑ 2
i=0 iβi
)
= −γ + 94 β − α
C3 = 13!
(∑ 2
i=0 i3αi
)
− 12!
(∑ 2
i=0 i2βi
)
= 74 β − 1
3 − γ3 − 3
2 α
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
Order of precision: LTE = C0y(tn) + C1hy ′(tn) + C2h2y ′′(tn) + ... + Cqhqy (q)(tn + ...
where:
C0 =∑ k
i=0 αi
C1 =∑ k
i=0 iαi −∑ k
i=0 βi
Cq = 1q!
(∑ k
i=0 iqαi
)
− 1(q−1)!
(∑ k
i=0 iq−1βi
)
, q ≥ 2
C0 =∑ 2
i=0 αi = 0C1 =
∑ 2i=0 iαi −
∑ 2i=0 βi = −2γ + 3
2 β
C2 = 12!
(∑ 2
i=0 i2αi
)
−(∑ 2
i=0 iβi
)
= −γ + 94 β − α
C3 = 13!
(∑ 2
i=0 i3αi
)
− 12!
(∑ 2
i=0 i2βi
)
= 74 β − 1
3 − γ3 − 3
2 α
The method is of order 2: α = 32 β = 2γ
Error constant: C = −2−3α6
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
Order of precision: LTE = C0y(tn) + C1hy ′(tn) + C2h2y ′′(tn) + ... + Cqhqy (q)(tn + ...
where:
C0 =∑ k
i=0 αi
C1 =∑ k
i=0 iαi −∑ k
i=0 βi
Cq = 1q!
(∑ k
i=0 iqαi
)
− 1(q−1)!
(∑ k
i=0 iq−1βi
)
, q ≥ 2
C0 =∑ 2
i=0 αi = 0C1 =
∑ 2i=0 iαi −
∑ 2i=0 βi = −2γ + 3
2 β
C2 = 12!
(∑ 2
i=0 i2αi
)
−(∑ 2
i=0 iβi
)
= −γ + 94 β − α
C3 = 13!
(∑ 2
i=0 i3αi
)
− 12!
(∑ 2
i=0 i2βi
)
= 74 β − 1
3 − γ3 − 3
2 α
The method is of order 2: α = 32 β = 2γ
Error constant: C = −2−3α6
Second order BDF-α:(
3
2+ α
)
yn+2 + (−2 − 2α) yn+1 +
(1
2+ α
)
yn = h(1 + α)fn+2 − hαfn+1
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
Order of precision: LTE = C0y(tn) + C1hy ′(tn) + C2h2y ′′(tn) + ... + Cqhqy (q)(tn + ...
where:
C0 =∑ k
i=0 αi
C1 =∑ k
i=0 iαi −∑ k
i=0 βi
Cq = 1q!
(∑ k
i=0 iqαi
)
− 1(q−1)!
(∑ k
i=0 iq−1βi
)
, q ≥ 2
C0 =∑ 2
i=0 αi = 0C1 =
∑ 2i=0 iαi −
∑ 2i=0 βi = −2γ + 3
2 β
C2 = 12!
(∑ 2
i=0 i2αi
)
−(∑ 2
i=0 iβi
)
= −γ + 94 β − α
C3 = 13!
(∑ 2
i=0 i3αi
)
− 12!
(∑ 2
i=0 i2βi
)
= 74 β − 1
3 − γ3 − 3
2 α
The method is of order 2: α = 32 β = 2γ
Error constant: C = −2−3α6
Second order BDF-α:(
3
2+ α
)
yn+2 + (−2 − 2α) yn+1 +
(1
2+ α
)
yn = h(1 + α)fn+2 − hαfn+1
Cases:
α = −0.5 ⇒ Trapezoidal method
α = 0 ⇒ BDF2 method
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability regions
After applying the method to the test equation:(
32 + α
)yn+2 + (−2 − 2α) yn+1 +
(12 + α
)yn = h(1 + α)yn+2 − hαyn+1
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability regions
After applying the method to the test equation:(
32 + α
)yn+2 + (−2 − 2α) yn+1 +
(12 + α
)yn = h(1 + α)yn+2 − hαyn+1
Frontier: h =
(32 +α
)r2+(−2−2α)r+
(12 +α
)
(1+α)r2−αr
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability regions
After applying the method to the test equation:(
32 + α
)yn+2 + (−2 − 2α) yn+1 +
(12 + α
)yn = h(1 + α)yn+2 − hαyn+1
Frontier: h =
(32 +α
)r2+(−2−2α)r+
(12 +α
)
(1+α)r2−αr
After substituting r = eiθ : h(θ) =(1+2α)(cosθ−1)2+isinθ
[(1+2α)(1−cosθ)+ 1
1+α
]
(1+α)
[(cosθ−
α1+α
)2+sin2θ
]
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability regions
After applying the method to the test equation:(
32 + α
)yn+2 + (−2 − 2α) yn+1 +
(12 + α
)yn = h(1 + α)yn+2 − hαyn+1
Frontier: h =
(32 +α
)r2+(−2−2α)r+
(12 +α
)
(1+α)r2−αr
After substituting r = eiθ : h(θ) =(1+2α)(cosθ−1)2+isinθ
[(1+2α)(1−cosθ)+ 1
1+α
]
(1+α)
[(cosθ−
α1+α
)2+sin2θ
]
For α ≥ −0.5 the denominator of h(θ) is lower bounded. Fixing α ≥ −0.5, for a sufficientlybig real number which depends on α and independent of θ, R (α) ∈ R, the real part h(θ)verifies: 0 ≤ Re(h(θ)) ≤ R(α)
The frontier of the stability region h(θ) lies inthe right semiplane C
+.For h ∈ C
−, A-stability is achieved and usingcontinuity, C
− belongs to the stability region.A-stable when α ∈ [−0.5, +∞)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Stability regions
After applying the method to the test equation:(
32 + α
)yn+2 + (−2 − 2α) yn+1 +
(12 + α
)yn = h(1 + α)yn+2 − hαyn+1
Frontier: h =
(32 +α
)r2+(−2−2α)r+
(12 +α
)
(1+α)r2−αr
After substituting r = eiθ : h(θ) =(1+2α)(cosθ−1)2+isinθ
[(1+2α)(1−cosθ)+ 1
1+α
]
(1+α)
[(cosθ−
α1+α
)2+sin2θ
]
For α ≥ −0.5 the denominator of h(θ) is lower bounded. Fixing α ≥ −0.5, for a sufficientlybig real number which depends on α and independent of θ, R (α) ∈ R, the real part h(θ)verifies: 0 ≤ Re(h(θ)) ≤ R(α)
The frontier of the stability region h(θ) lies inthe right semiplane C
+.For h ∈ C
−, A-stability is achieved and usingcontinuity, C
− belongs to the stability region.A-stable when α ∈ [−0.5, +∞)
−2 0 2 4 6 8 10 12 14−8i
−6i
−4i
−2i
0
2i
4i
6i
8i
α=−0.4
α=−0.3
α=−0.2
α=0
α=4
α=100
α=1
++++++
α=−0.1
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
The expression obtained after applying the method to the test equation in matrix form:Yn+2 = AYn+1
where:
Yn+2 = (yn+1, yn+2)T , Yn+1 = (yn, yn+1)
T , A = A−11 A2
A1 =
(1 00 3
2 + α − h(1 + α)
)
, A2 =
(0 1
− 12 − α 2 + 2α − hα
)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
The expression obtained after applying the method to the test equation in matrix form:Yn+2 = AYn+1
where:
Yn+2 = (yn+1, yn+2)T , Yn+1 = (yn, yn+1)
T , A = A−11 A2
A1 =
(1 00 3
2 + α − h(1 + α)
)
, A2 =
(0 1
− 12 − α 2 + 2α − hα
)
Eigenvalues of the amplification matrix:
λ1,2 =−2 − 2α + hα ±
√
h2α2 + 2h(α + 1) + 1
−3 − 2α + 2h(1 + α)(1)
To characterize the numerical dissipation, the espectral radius when h → ∞ is calculated. Forthe A-stable BDF-α, that is to say, α ∈ [−0.5, +∞), we obtain:
ρ∞ =
1, α = −0.5−2α2+2α
< 1, α ∈ [−0.5, 0)2α
2+2α< 1, α ∈ [0, +∞)
Which means that fixing α ∈ [−0.5, +∞) ρ∞ takes all the values of (0, 1].
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Considerations about the new method
10−2
10−1
100
101
102
103
104
0
0.2
0.4
0.6
0.8
1
Ω/(2π)
ρ
Trapezoidal
BDF−α=9.50
Collocation
(γ=0.5,β=0.16,θ=1.514951)
BDF−α=0Houbolt
BDF−α=1.17
HHT−α (α= 0.05)
BDF−α=−0.35
HHT−α (α= 0.3)
BDF−α=−0.475065
10−0.8
10−0.6
10−0.4
10−0.2
100
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
Ω/(2π)
ρ
Trapezoidal
HHT−α (α= 0.05)
BDF−α=−0.475065
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Formulation of the FEM approximation of the Laplace-Poisson non-linear PDE:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Formulation of the FEM approximation of the Laplace-Poisson non-linear PDE:
Strong formulation:φ = −k(u)∇u. Find u(x) ∈ C2(Ω) which verifies:
∇ · φ(u(x)) = f (x), ∀x ∈ Ωu(x) = g(x), ∀x ∈ Γg
φ(u(x)) · n = −h(x), ∀x ∈ Γh
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Formulation of the FEM approximation of the Laplace-Poisson non-linear PDE:
Strong formulation:φ = −k(u)∇u. Find u(x) ∈ C2(Ω) which verifies:
∇ · φ(u(x)) = f (x), ∀x ∈ Ωu(x) = g(x), ∀x ∈ Γg
φ(u(x)) · n = −h(x), ∀x ∈ Γh
Weak formulation: ∫
Γhw(x) (φ(u(x)) · n) dΓh −
∫
Ω∇w(x) · φ(u(x))dΩ =
∫
Ωw(x)f (x)dΩ, ∀w(x) ∈ V
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Formulation of the FEM approximation of the Laplace-Poisson non-linear PDE:
Strong formulation:φ = −k(u)∇u. Find u(x) ∈ C2(Ω) which verifies:
∇ · φ(u(x)) = f (x), ∀x ∈ Ωu(x) = g(x), ∀x ∈ Γg
φ(u(x)) · n = −h(x), ∀x ∈ Γh
Weak formulation: ∫
Γhw(x) (φ(u(x)) · n) dΓh −
∫
Ω∇w(x) · φ(u(x))dΩ =
∫
Ωw(x)f (x)dΩ, ∀w(x) ∈ V
FEM approximation: uh(x) =∑
j∈η
uj Nj (x)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Formulation of the FEM approximation of the Laplace-Poisson non-linear PDE:
Strong formulation:φ = −k(u)∇u. Find u(x) ∈ C2(Ω) which verifies:
∇ · φ(u(x)) = f (x), ∀x ∈ Ωu(x) = g(x), ∀x ∈ Γg
φ(u(x)) · n = −h(x), ∀x ∈ Γh
Weak formulation: ∫
Γhw(x) (φ(u(x)) · n) dΓh −
∫
Ω∇w(x) · φ(u(x))dΩ =
∫
Ωw(x)f (x)dΩ, ∀w(x) ∈ V
FEM approximation: uh(x) =∑
j∈η
uj Nj (x)
Matricial format: uh(x) =
n∑
j=1
uj Nj (x) ⇒ uh(x) = N · U = (N1(x), ..., Nn(x))
︸ ︷︷ ︸
N
·
u1
...un
︸ ︷︷ ︸
U
= N · U
∇uh(x) =∑ n
j=1 uj∇Nj (x) ⇒ ∇uh(x) =
(∂N1∂x · · · ∂Nn
∂x∂N1∂y · · · ∂Nn
∂y
)
︸ ︷︷ ︸
B
·
u1
...un
︸ ︷︷ ︸
U
= B · U
φ = −k(u)BU
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Formulation of the FEM approximation of the Laplace-Poisson non-linear PDE:
Strong formulation:φ = −k(u)∇u. Find u(x) ∈ C2(Ω) which verifies:
∇ · φ(u(x)) = f (x), ∀x ∈ Ωu(x) = g(x), ∀x ∈ Γg
φ(u(x)) · n = −h(x), ∀x ∈ Γh
Weak formulation: ∫
Γhw(x) (φ(u(x)) · n) dΓh −
∫
Ω∇w(x) · φ(u(x))dΩ =
∫
Ωw(x)f (x)dΩ, ∀w(x) ∈ V
FEM approximation: uh(x) =∑
j∈η
uj Nj (x)
Matricial format: uh(x) =
n∑
j=1
uj Nj (x) ⇒ uh(x) = N · U = (N1(x), ..., Nn(x))
︸ ︷︷ ︸
N
·
u1
...un
︸ ︷︷ ︸
U
= N · U
∇uh(x) =∑ n
j=1 uj∇Nj (x) ⇒ ∇uh(x) =
(∂N1∂x · · · ∂Nn
∂x∂N1∂y · · · ∂Nn
∂y
)
︸ ︷︷ ︸
B
·
u1
...un
︸ ︷︷ ︸
U
= B · U
φ = −k(u)BU
Non-linear equations system: Fint (U) = Fext
where:
Fint (U) = −
∫
Ωh BTa φh(U)dΩh, donde φh = −k(u)∇uh(x) = −k(u)BU
Fext =∫
Ωh NTa f (x)dΩh +
∫
Γhh
NTa h(x)dΓh
h
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
Formulation of the FEM approximation of the Laplace-Poisson non-linear PDE:
Strong formulation:φ = −k(u)∇u. Find u(x) ∈ C2(Ω) which verifies:
∇ · φ(u(x)) = f (x), ∀x ∈ Ωu(x) = g(x), ∀x ∈ Γg
φ(u(x)) · n = −h(x), ∀x ∈ Γh
Weak formulation: ∫
Γhw(x) (φ(u(x)) · n) dΓh −
∫
Ω∇w(x) · φ(u(x))dΩ =
∫
Ωw(x)f (x)dΩ, ∀w(x) ∈ V
FEM approximation: uh(x) =∑
j∈η
uj Nj (x)
Matricial format: uh(x) =
n∑
j=1
uj Nj (x) ⇒ uh(x) = N · U = (N1(x), ..., Nn(x))
︸ ︷︷ ︸
N
·
u1
...un
︸ ︷︷ ︸
U
= N · U
∇uh(x) =∑ n
j=1 uj∇Nj (x) ⇒ ∇uh(x) =
(∂N1∂x · · · ∂Nn
∂x∂N1∂y · · · ∂Nn
∂y
)
︸ ︷︷ ︸
B
·
u1
...un
︸ ︷︷ ︸
U
= B · U
φ = −k(u)BU
Non-linear equations system: Fint (U) = Fext
where:
Fint (U) = −
∫
Ωh BTa φh(U)dΩh, donde φh = −k(u)∇uh(x) = −k(u)BU
Fext =∫
Ωh NTa f (x)dΩh +
∫
Γhh
NTa h(x)dΓh
h
Newton Raphson resolution:
R(Uk ) = Fext − Fint (Uk )
Uk+1 = Uk − J−1R(Uk )k = 0,1,2,... until ‖R (Uk+1)‖ < tol ≈ 0 J(Uk ) = −K t (Uk ) =
∂R(Uk )
∂Uk= −
∂Fkint
(Uk )
∂Uk
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
The dynamic problem:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
The dynamic problem:
FEM approximation: uh(x) = N(x)U(t)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
The dynamic problem:
FEM approximation: uh(x) = N(x)U(t)
Function f (x) of the anterior section becomes:
f(x, t) − ρutt (x, t)︸ ︷︷ ︸
inertia forces
− cut (x, t)︸ ︷︷ ︸
damping forces
≈ f(x, t) − ρN(x)U′′(t) − cN(x)U′
(t)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
The dynamic problem:
FEM approximation: uh(x) = N(x)U(t)
Function f (x) of the anterior section becomes:
f(x, t) − ρutt (x, t)︸ ︷︷ ︸
inertia forces
− cut (x, t)︸ ︷︷ ︸
damping forces
≈ f(x, t) − ρN(x)U′′(t) − cN(x)U′
(t)
Second order ODE system: MU′′ + CU′ + Fint (U) = Fext
where:
M =∫
ΩNT ρN dΩ , C =
∫
ΩNT cN dΩ
Fint =∫
ΩBT σh(U) dΩ , Fext =
∫
ΓtNT t dSt +
∫
ΩNT f dΩ
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
The dynamic problem:
FEM approximation: uh(x) = N(x)U(t)
Function f (x) of the anterior section becomes:
f(x, t) − ρutt (x, t)︸ ︷︷ ︸
inertia forces
− cut (x, t)︸ ︷︷ ︸
damping forces
≈ f(x, t) − ρN(x)U′′(t) − cN(x)U′
(t)
Second order ODE system: MU′′ + CU′ + Fint (U) = Fext
where:
M =∫
ΩNT ρN dΩ , C =
∫
ΩNT cN dΩ
Fint =∫
ΩBT σh(U) dΩ , Fext =
∫
ΓtNT t dSt +
∫
ΩNT f dΩ
If the problem is linear
Fint = KU, where K =∫
ΩBT kBU dΩ
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology
The dynamic problem:
FEM approximation: uh(x) = N(x)U(t)
Function f (x) of the anterior section becomes:
f(x, t) − ρutt (x, t)︸ ︷︷ ︸
inertia forces
− cut (x, t)︸ ︷︷ ︸
damping forces
≈ f(x, t) − ρN(x)U′′(t) − cN(x)U′
(t)
Second order ODE system: MU′′ + CU′ + Fint (U) = Fext
where:
M =∫
ΩNT ρN dΩ , C =
∫
ΩNT cN dΩ
Fint =∫
ΩBT σh(U) dΩ , Fext =
∫
ΓtNT t dSt +
∫
ΩNT f dΩ
If the problem is linear
Fint = KU, where K =∫
ΩBT kBU dΩ
First order equivalent ODE system:
d(t) = U(t)v(t) = U′(t) ⇒
d ′(t) = U′(t) = v(t)
Mv ′(t) = MU′′(t) = −Fint (d) − Cv + Fext ⇒
(I 00 M
)
︸ ︷︷ ︸
M∗
(d ′(t)v ′(t)
)
︸ ︷︷ ︸
y′
=
(v(t)
R(d) − Cv
)
︸ ︷︷ ︸
f (t,y)
⇒ M∗y ′= f (t, y)
where:
f1(t, y) ≡ y2 = v , f2(t, y) ≡ R(y1) − Cy2 = R(d) − Cv
j =∂f (t,y)
∂y =
∂f1∂y1
∂f1∂y2
∂f2∂y1
∂f2∂y2
=
( ∂v∂d
∂v∂v
∂(R(d)−Cv)∂d
∂(R(d)−Cv)∂v
)
=
(0 I
∂R(d)∂d −C
)
=
(0 I
J(U) −C
)
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object Oriented Programming methodology(OOP)
Resulting OOP architecture: 3 blocks
Mesh object (“objmalla”): Related to the calculus of the domain integrals.Problem object (“objprob”): Related to the management of the specific aspects of the problem.Method object (“objmetodo”): Related to the resolution of the problem (linear, non-linear,dynamic).
objmef
objelef
objelep
objfty
objpint
objdef
objprobobjmetodo
objnr
objJ
objmalla
objresYcc objmat objU
objsislin
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode)
The step by step advancing process of the different methods has a common estructure:initialization, loop in steps, actualization.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode)
Trapezoidal rule → trapezoidal object:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode)
Trapezoidal rule → trapezoidal object:
Non-linear problem:
M·y ′ = f (t, y) ⇒ M · yn+1 = M · yn + h2 (fn + fn+1) Each iteration is solved by Mewton-Raphson:
ym+1n+1 = ym
n+1 −(J(ym
n+1))−1 R
(ym
n+1
), m = 0, 1, 2, ... paso(step)
where:
R(ym
n+1
)= 2
h M(ym
n+1 − yn)−(fn + f m
n+1
)calcR
J(ym
n+1
)=
∂R(ymn+1)
∂yn+1= 2
h M − jmn+1 calcJ
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode)
Trapezoidal rule → trapezoidal object:
Non-linear problem:
M·y ′ = f (t, y) ⇒ M · yn+1 = M · yn + h2 (fn + fn+1) Each iteration is solved by Mewton-Raphson:
ym+1n+1 = ym
n+1 −(J(ym
n+1))−1 R
(ym
n+1
), m = 0, 1, 2, ... paso(step)
where:
R(ym
n+1
)= 2
h M(ym
n+1 − yn)−(fn + f m
n+1
)calcR
J(ym
n+1
)=
∂R(ymn+1)
∂yn+1= 2
h M − jmn+1 calcJ
Linear problem:
M·y ′ = A · y + g(t) ⇒ where:
First order Laplace: A → −K , y → d(t), g(t) → Fext (t)
Wave: M →
(I 00 M
)
, A →
(0 I
−K −C
)
, y →
(d(t)v(t)
)
, g(t) →
(0
Fext (t)
)
Generic case: M, A, g(t)
Advancing formula:
Myn+1 = Myn + h2 (Ayn + g(tn) + Ayn+1 + g(tn+1)) ⇒
(M − h
2 A)
yn+1 =(M + h
2 A)
yn + h2 (g(tn) + g(tn+1))
yn+1 = C1
[
C2yn +h
2(g(tn) + g(tn+1))
]
where: C1 =(M − h
2 A)−1, C2 =
(M + h
2 A). C1, C2, M and A are calculated once in the initialization.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode): BDF method
Multistep methods → BDFs:
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode): BDF method
Multistep methods → BDFs:
Non-linear problem:
∑ kj=0 αj Myn+j = hfn+k ⇒
R(ym
n+k
)= 1
h
(
αk Mymn+k +
∑ k−1j=0 αj Myn+j
)
− f mn+k
J(ym
n+k
)=
∂R(ymn+k
)
∂yn+k= 1
h αk M − jmn+k
Programming:
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− f m
n+kJ(ym
n+k
)= 1
h αk M − jmn+k
where:
C1 = (αk M − hA)−1 , only for the linear caseC2 =
(αk−1M αk−2M αk−3M . . . α1M α0M
)
Yn+k−1 =(yn+k−1 yn+k−2 yn+k−3 . . . yn+1 yn
)T
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode): BDF method
Multistep methods → BDFs:
Non-linear problem:
∑ kj=0 αj Myn+j = hfn+k ⇒
R(ym
n+k
)= 1
h
(
αk Mymn+k +
∑ k−1j=0 αj Myn+j
)
− f mn+k
J(ym
n+k
)=
∂R(ymn+k
)
∂yn+k= 1
h αk M − jmn+k
Programming:
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− f m
n+kJ(ym
n+k
)= 1
h αk M − jmn+k
where:
C1 = (αk M − hA)−1 , only for the linear caseC2 =
(αk−1M αk−2M αk−3M . . . α1M α0M
)
Yn+k−1 =(yn+k−1 yn+k−2 yn+k−3 . . . yn+1 yn
)T
Linear problem:∑ k
j=0 αj Myn+j = h (Ayn+k + g(tn+k )) ⇒ (αk M − hA) yn+k = −∑ k−1
j=0 αj Myn+j + hg(tn+k )
(αk M − hA) yn+k = −C2Yn+k−1 + hg(tn+k )
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode): superfuturepoints methods
Superfuture points methods → EBDFs:
Non-linear problem:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− βk f m
n+k − βk+1 fn+k+1
J(ym
n+k
)= 1
h αk M − βk jmn+k
where:
C2 =(αk−1M αk−2M αk−3M . . . α1M α0M
)
Yn+k−1 =(yn+k−1 yn+k−2 yn+k−3 . . . yn+1 yn
)T
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode): superfuturepoints methods
Superfuture points methods → EBDFs:
Non-linear problem:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− βk f m
n+k − βk+1 fn+k+1
J(ym
n+k
)= 1
h αk M − βk jmn+k
where:
C2 =(αk−1M αk−2M αk−3M . . . α1M α0M
)
Yn+k−1 =(yn+k−1 yn+k−2 yn+k−3 . . . yn+1 yn
)T
Superfuture points methods → MEBDFs:
Non-linear problem:
∑ kj=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1 + h
(
βk − βk
)
fn+k
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− βk f m
n+k − βk+1 fn+k+1 −(
βk − βk
)
fn+k
J(ym
n+k
)=
∂R(ymn+k
)
∂yn+k= 1
h αk M − βk jmn+k
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Object method ODE (objode): superfuturepoints methods
Superfuture points methods → EBDFs:
Non-linear problem:∑ k
j=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− βk f m
n+k − βk+1 fn+k+1
J(ym
n+k
)= 1
h αk M − βk jmn+k
where:
C2 =(αk−1M αk−2M αk−3M . . . α1M α0M
)
Yn+k−1 =(yn+k−1 yn+k−2 yn+k−3 . . . yn+1 yn
)T
Superfuture points methods → MEBDFs:
Non-linear problem:
∑ kj=0 αj yn+j = hβk fn+k + hβk+1 fn+k+1 + h
(
βk − βk
)
fn+k
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− βk f m
n+k − βk+1 fn+k+1 −(
βk − βk
)
fn+k
J(ym
n+k
)=
∂R(ymn+k
)
∂yn+k= 1
h αk M − βk jmn+k
Programming both:
R(ym
n+k
)= 1
h
(αk Mym
n+k + C2Yn+k−1)− γk f m
n+k − βk+1 fn+k+1 − γk fn+k
J(ym
n+k
)=
∂R(ymn+k
)
∂yn+k= 1
h αk M − γk jmn+k
where:For EBDFs: γk = βk and γk = 0.
For MEBDFs: γk = βk and γk =(
βk − βk
)
.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 1: Linear wave equation with rectangular pulse IC (1D).400 elements and 1400 steps.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 1: Linear wave equation with rectangular pulse IC (1D).400 elements and 1400 steps.
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método HHT−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=0.3 γ=0.8 β=0.4225
t= 0t= 2
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 1: Linear wave equation with rectangular pulse IC (1D).400 elements and 1400 steps.
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método HHT−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=0.3 γ=0.8 β=0.4225
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método BDF−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=−0.35
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 1: Linear wave equation with rectangular pulse IC (1D).400 elements and 1400 steps.
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método HHT−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=0.3 γ=0.8 β=0.4225
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método BDF−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=−0.35
Computational times → HHT-α: 1.88 seconds, BDF-α: 6.17 secondsBoth very quick but HHT-α 3 times quicker.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 1: Linear wave equation with rectangular pulse IC (1D).400 elements and 1400 steps.
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método HHT−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=0.3 γ=0.8 β=0.4225
t= 0t= 2
0 1 2 3 4 5 6 7 8−0.5
0
0.5
1
1.5Método BDF−alfa, tiempo=16, nele=400, pasos=1400, masa=cons
α=−0.35
Computational times → HHT-α: 1.88 seconds, BDF-α: 6.17 secondsBoth very quick but HHT-α 3 times quicker.The most expensive operations:- HHT-α: an+1 = MCK−1 · Fn+1
where:
MCK = (M + (1 − α)Chγ + (1 − α)Kh2β)
Fn+1 = Fext (tn+1−α) − C ((1 − α)vn+1 + αvn) − K(
(1 − α)dn+1 + αdn
)
- BDF-α: yn+2 = C−11 · [−C2Yn+1 − hαAyn+1 + h(1 + α)g(tn+2) − hαg(tn+1)]
︸ ︷︷ ︸
TI
⇒ yn+2 = C1 · TI
where:
C1 = (β2M − h(1 + α)A) ,
C2 = (β1M β0M), Yn+1 = (yn+1 yn)T
The dimension of the matrix C1 = (β2M − h(1 + α)A) is the double of MCK .
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 2: Non-linear PDE of a guitar string.20 elements and 1000 steps.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 2: Non-linear PDE of a guitar string.20 elements and 1000 steps.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.015198, nele=20, pasos=1000, masa=cons
α=0.3 , γ=0.8, β=0.4225
Figure: HHT-α = 0.3.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 2: Non-linear PDE of a guitar string.20 elements and 1000 steps.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.015198, nele=20, pasos=1000, masa=cons
α=0.3 , γ=0.8, β=0.4225
Figure: HHT-α = 0.3.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
α=−0.35
Método BDF−α, tiempo=0.015198, nele=20, pasos=1000, masa=cons
Figure: BDF-α = −0.35.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 2: Non-linear PDE of a guitar string.20 elements and 1000 steps.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.015198, nele=20, pasos=1000, masa=cons
α=0.3 , γ=0.8, β=0.4225
Figure: HHT-α = 0.3.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
α=−0.35
Método BDF−α, tiempo=0.015198, nele=20, pasos=1000, masa=cons
Figure: BDF-α = −0.35.
Computational times → HHT-α = 0.3: 244.76 seconds, 9479 iterations. BDF-α = −0.35:259.89 seconds, 9516 iterations.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 2: Non-linear PDE of a guitar string.20 elements and 1000 steps.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08Método HHT−alfa, tiempo=0.015198, nele=20, pasos=1000, masa=cons
α=0.3 , γ=0.8, β=0.4225
Figure: HHT-α = 0.3.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
α=−0.35
Método BDF−α, tiempo=0.015198, nele=20, pasos=1000, masa=cons
Figure: BDF-α = −0.35.
Computational times → HHT-α = 0.3: 244.76 seconds, 9479 iterations. BDF-α = −0.35:259.89 seconds, 9516 iterations.Again, the dimension of the matrices of the BDF-α method is the double.Time for solving the equation system of the total iterations→ HHT-α: 1.54 seconds andBDF-α 0.1 seconds.This difference is not important in the final balance as it is the calculation of R and J of themethods which more time consumes.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D.
A plane deformation square block of 2 units.Data: E = 1000, ν = 0.25, ρ = 1, IC that corresponds to a vertical lengthening of 1 unit, BCDirichlet nule for the verticals of the horizontal simmetry axis and for the horizontaldisplacement of the vertical simmetry axis.Only the forth part of the block has been discretized due to the symmetry.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D.
A plane deformation square block of 2 units.Data: E = 1000, ν = 0.25, ρ = 1, IC that corresponds to a vertical lengthening of 1 unit, BCDirichlet nule for the verticals of the horizontal simmetry axis and for the horizontaldisplacement of the vertical simmetry axis.Only the forth part of the block has been discretized due to the symmetry.
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
tensiones: yy
−300
−200
−100
0
100
200
300
400
500
Figure: Final stresses anddeformation (t = 0.5).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D.
A plane deformation square block of 2 units.Data: E = 1000, ν = 0.25, ρ = 1, IC that corresponds to a vertical lengthening of 1 unit, BCDirichlet nule for the verticals of the horizontal simmetry axis and for the horizontaldisplacement of the vertical simmetry axis.Only the forth part of the block has been discretized due to the symmetry.
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
tensiones: yy
−300
−200
−100
0
100
200
300
400
500
Figure: Final stresses anddeformation (t = 0.5).
0 0.1 0.2 0.3 0.4 0.5−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Figure: Nodal displacements(Trap. method 200 steps).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D → Harmonic response
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D → Harmonic response
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4Modo nº 1 de 60 ; Frec: w=44.7771 ; Partic: F=0.33037
0 0.1 0.2 0.3 0.4 0.5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
Figure: First mode deformation as IC (left) and nodal displacements (right) (Trap. 200steps).
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D → Harmonic response
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4Modo nº 1 de 60 ; Frec: w=44.7771 ; Partic: F=0.33037
0 0.1 0.2 0.3 0.4 0.5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
Figure: First mode deformation as IC (left) and nodal displacements (right) (Trap. 200steps).
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4Modo nº 60 de 60, Frec: w=680.1678 ; Partic: F=3.3282e−005
0 0.01 0.02 0.03 0.04 0.05−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
Figure: Last mode deformation as IC (left) and nodal displacements (right) (Trap. 1000
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D.
0 0.5 1 1.5 2−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1BDF−α=−0.35, 317 pasos
0 0.5 1 1.5 2−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1BDF−α=−0.35, 1000 pasos
0 0.5 1 1.5 2−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1BDF−α=−0.35, 2000 pasos
Figure: Algorithmic damping for high frequency, BDF-α = −0.35 (317, 1000 and 2000steps). Last mode deformation.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Results
Example 3: Dynamic linear elasticity 2D.
0 0.5 1 1.5 2−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1BDF−α=−0.35, 317 pasos
0 0.5 1 1.5 2−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1BDF−α=−0.35, 1000 pasos
0 0.5 1 1.5 2−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1BDF−α=−0.35, 2000 pasos
Figure: Algorithmic damping for high frequency, BDF-α = −0.35 (317, 1000 and 2000steps). Last mode deformation.
0 0.5 1 1.5 2−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1BDF−α=−0.35, 317 pasos, evolución del primer modo
Figure: First mode evolution, BDF-α = −0.35 (317steps).
The solution does not lose too muchprecision.
Numericalmethods forstiff ODEs
ElisabeteAlberdi Celaya
Introduction
First orderODEs
ChangingEBDFs andMEBDFs
LMS forsecond orderODEs
BDF-αmethod
OOPmethodology
Results
Thank You foryour attention
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