Numerical stiff ODEs Numerical methods for stiff Ordinary ... Alberdi Celaya Introduction First...
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Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
Numerical methods for stiff Ordinary differential equations. Application to the
Finite Element Method (FEM)
Elisabete Alberdi Celaya
EUIT de Minas y Obras Públicas UPV/EHU, Paseo Rafael Moreno Pitxitxi 2, 48013 Bilbao (Vizcaya)
April 7, 2013
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
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
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
PDEs→ FEM approximation
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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]
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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]
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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
N i (x
j )=δ
ij
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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
N i (x
j )=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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
N i (x
j )=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)
Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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
N i (x
j )=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=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 = ∫ L
0 N ′
i ku h x dx, i = 2, ..., n − 1
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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
N i (x
j )=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=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 = ∫ L
0 N ′
i ku h x dx, i = 2, ..., n − 1
Ordinary Differential Equations System:
∑ n−1 j=2
∫ L
0 ρcpNi (x)Nj (x)dx
︸ ︷︷ ︸
mij
d ′j (t) = − ∑ n−1
j=2
∫ L
0 kN′i (x)N
′
j (x)dx
︸ ︷︷ ︸
kij
dj (t), i, j = 2, ..., n − 1
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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
N i (x
j )=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=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 = ∫ L
0 N ′
i ku h x dx, i = 2, ..., n − 1
Ordinary Differential Equations System:
∑ n−1 j=2
∫ L
0 ρcpNi (x)Nj (x)dx
︸ ︷︷ ︸
mij
d ′j (t) = − ∑ n−1
j=2
∫ L
0 kN′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
Numerical methods for stiff ODEs
Elisabete Alberdi Celaya
Introduction
First order ODEs
Changing EBDFs and MEBDFs
LMS for second order ODEs
BDF-α method
OOP methodology
Results
A FEM application to the 1D linear diffusion equation
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
N i (x
j )=δ
ij
FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=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 = ∫ L
0 N ′
i ku h x dx, i = 2, ..., n − 1
Ordinary Differential Equations System:
∑ n−1 j=2
∫ L
0 ρcpNi (x)Nj (x)dx
︸ ︷︷ ︸
mij
d ′j (t) = − ∑ n−1
j=2
∫ L
0 kN′i (x)N
′
j (x)dx
︸ ︷︷ ︸
kij
dj (t), i, j = 2, ..., n − 1
DIFFUSION EQUA