The integration of stiff systems of ODEs using multistep methods … · The integration of stiff...

Numericalmethods forstiff ODEs

ElisabeteAlberdi

Celaya1 , JuanJose Anza2

Introduction

LMS forsecond orderODEs

First orderODEs

BDF-αmethod

Results

Conclusions

The integration of stiff systems of ODEsusing multistep methods

Elisabete Alberdi Celaya1, Juan Jose Anza2

1Department of Applied Mathematics, EUIT de Minas y Obras Publicas,2Department of Applied Mathematics, ETS de Ingenierıa de Bilbao,

1,2University of the Basque Country UPV/EHU, Bilbao (Spain)

December 10, 2013

2Sm


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Index

1 Introduction

2 Linear multistep methods for 2nd order ODEs

3 Numerical methods for first order ODEs

4 BDF-α method

5 Results

6 Conclusions


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

A FEM application to the 1D linear diffusionand wave equation

PDEs→ FEM approximation


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0

0

1

Ni(x

j)=δ

ij


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0

0

1

Ni(x

j)=δ

ij

FEM approximation: u(x, t) ≈ uh(x, t) =∑ n−1

j=2 dj (t)Nj (x)


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0

0

1

Ni(x

j)=δ

ij


j=2 dj (t)Nj (x)

Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0

0

1

Ni(x

j)=δ

ij


j=2 dj (t)Nj (x)


Weak formulation: ∫ L0 Ni ρcput dx =

∫ L0 N′

i kuhx dx, i = 2, ..., n − 1


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0

0

1

Ni(x

j)=δ

ij


j=2 dj (t)Nj (x)



∫ 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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0

0

1

Ni(x

j)=δ

ij


j=2 dj (t)Nj (x)



∫ L0 N′

i kuhx dx, i = 2, ..., n − 1


∑ 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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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]

Wave:

utt = α2uxx

CC : u(0, t) = 0 = u(L, t)

CI : u(x , 0) = g(x), ut(x , 0) = 0

0

1

Ni(x

j)=δ

ij


j=2 dj (t)Nj (x)



∫ L0 N′

i kuhx dx, i = 2, ..., n − 1


∑ 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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Linear wave equation examples in MATLAB

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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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



∞

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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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



∞

k=1 Ak sin(

kπx8

)cos(ωk t), where:

ωk = kπ

8 , φk = sin(

kπx8

)

Ak = 2L

∫ L0 g(x) sin

(kπx

8

)dx


Md′′(t) = −K d(t) ⇒ u(x, t) ≈ uh(x, t) =∑ n−2


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.


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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



∞

k=1 Ak sin(

kπx8

)cos(ωk t), where:

ωk = kπ

8 , φk = sin(

kπx8

)

Ak = 2L

∫ L0 g(x) sin

(kπx

8

)dx


Md′′(t) = −K d(t) ⇒ u(x, t) ≈ uh(x, t) =∑ n−2


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).


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


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.


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


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).


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.51599 modos continuos

desplamiento nodos − tiempo


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1




ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1




ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

25 modos continuos



ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

25 modos continuos


DISCRETES


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

25 modos continuos


DISCRETES

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5Método supmod., nele=1600, nº modos=1599


t= 0t= 2


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

25 modos continuos


DISCRETES

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

25 modos continuos


DISCRETES

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

25 modos continuos


DISCRETES

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


CONTINUOUS

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

25 modos continuos


DISCRETES

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



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.


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Integration of the ODE system which comes from FEM.

- Matlab odesuite: ode45, ode15s. Adaptative step size.- Stiffness → existence of eigenvalues of different magnitude in the solution.- Stiffness, makes the solution expensive (more steps).- Increase of the number of elements, increases stiffness.

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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions




Wave equation:


T

d′(0) = d′

0 = (0, ..., 0))T


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.


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions




Wave equation:


T

d′(0) = d′

0 = (0, ..., 0))T


Senoidal:

0 1 2 3 4 5 6 7 8−1.5

−1

−0.5

0

0.5

1



t= 0t= 2t= 4t= 8t= 16


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


t= 0t= 2t= 4t= 8t= 10t= 16

The advantage of the ode15s disappears.


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions




Wave equation:


T

d′(0) = d′

0 = (0, ..., 0))T


Senoidal:

0 1 2 3 4 5 6 7 8−1.5

−1

−0.5

0

0.5

1



t= 0t= 2t= 4t= 8t= 16


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


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


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


400 elements:


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


400 elements:

Ode15s, 12837 steps:

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


ElisabeteAlberdi


Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


400 elements:


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



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


t= 0t= 2


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


400 elements:


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2

HHT-α method (“α” method),1400 steps:


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


400 elements:


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


400 elements:


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1


α=0.3 γ=0.8 β=0.4225

t= 0t= 2

Newmark’s method β = 1/6,γ = 0.5, 800 steps →Superconvergence:


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


400 elements:


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1



t= 0t= 2


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1


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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

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:


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



ρutt (x, t) =

T + E · S(√

1 + u2x (x, t) − 1

)

︸︷︷︸

T

uxx (x, t)



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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



ρutt (x, t) =

T + E · S(√

1 + u2x (x, t) − 1

)

︸︷︷︸

T

uxx (x, t)



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 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 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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



ρutt (x, t) =

T + E · S(√

1 + u2x (x, t) − 1

)

︸︷︷︸

T

uxx (x, t)



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 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 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


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 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



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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



ρutt (x, t) =

T + E · S(√

1 + u2x (x, t) − 1

)

︸︷︷︸

T

uxx (x, t)



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 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 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


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 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 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


α=0.3 , γ=0.8, β=0.4225


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

With this motivation the presentation is going to be about:

The study of the computational aspects of the MATLAB odesolver ode15s based onBackward Differentiation Formulae (BDF).

The study of the classical methods for second order ODEs of the mechanich which areable to dissipate the high-modes and a modification to second order BDF to obtain amethod with this feature.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Linear multistep methods for 2nd order ODEs

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.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Stiffness


- 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]


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Stiffness


- 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]

The stability and the numerical damping of the method: Apply the method to the second ordertest equation d ′′ + ω2d = 0, which represents an undamped vibrating physical system withnatural frequency f = ω/(2π) where w =

√

k/m:

Xn+1 = AXn (1)

where: Xn+i =(

dn+i , hvn+i , h2an+i

)Tfor i = 0, 1, h = ∆t and A is the amplification matrix.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Stiffness


- 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]


√

k/m:

Xn+1 = AXn (1)

where: Xn+i =(



Eigenvalues of matrix A are calculated and the largest one in module is the spectral radius:

ρ(A) = max |λi | : λi eigenvalue of A (2)


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Stiffness


- 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]


√

k/m:

Xn+1 = AXn (1)

where: Xn+i =(





The spectral radius is closely connected to the stability of the method and ρ(A) ≤ 1 isrequired. The method is unstable when γ < 1

2 and it is unconditionally stable when12 ≤ γ ≤ 2β.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Stiffness


- 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]


√

k/m:

Xn+1 = AXn (1)

where: Xn+i =(





The spectral radius is closely connected to the stability of the method and ρ(A) ≤ 1 isrequired. The method is unstable when γ < 1

2 and it is unconditionally stable when12 ≤ γ ≤ 2β.

High frequency dissipation is achieved when: β =

(γ+ 1

2

)2

4


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Newmark’s method can be reduced to a difference equation in the displacements, which takesthe form of a linear multistep method for second order differential equations:

2∑

i=0

αi dn+i = h22∑

i=0

βi d′′

n+i (3)

where the coefficients αj , βj are given by:

α0 = 1, β0 = −γ + β + 12

α1 = −2, β1 = −2β + γ + 12

α2 = 1, β2 = β

(4)


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



2∑

i=0

αi dn+i = h22∑

i=0

βi d′′

n+i (3)


α0 = 1, β0 = −γ + β + 12

α1 = −2, β1 = −2β + γ + 12

α2 = 1, β2 = β

(4)

Applying the order conditions for linear multistep methods, the method results second-orderaccurate when γ = 1/2.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



2∑

i=0

αi dn+i = h22∑

i=0

βi d′′

n+i (3)


α0 = 1, β0 = −γ + β + 12

α1 = −2, β1 = −2β + γ + 12

α2 = 1, β2 = β

(4)

Applying the order conditions for linear multistep methods, the method results second-orderaccurate when γ = 1/2.

Second-order accurate condition does not allow numerical dissipation

In the second-order accurate Newmark method (γ = 1/2), β ≥ 1/4 retains unconditionalstability. If in addition, high frequency dissipation is required, β = 1/4 has to be verified. Inthis case, Newmark’s method becomes the trapezoidal method, and high modes are notdamped as ρ∞ = 1.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

HHT-α method

It is a modification made to the Newmark method, with the aim of obtaining numericaldissipation in the high frequencies while retaining the order and stability conditions. Theexpression of the time-discrete equation of motion is modified with a new parameter α asfollows:

man+1 + cvn+1−α + kdn+1−α = f (tn+1−α)

dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]

vn+1 = vn + ∆t [(1 − γ) an + γan+1]

(5)

where:

dn+1−α = (1 + α) dn+1 − αdn

vn+1−α = (1 + α) vn+1 − αvn

tn+1−α = (1 + α) tn+1 − αtn(6)

If α = 0, the HHT-α method is reduced to Newmark’s method.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

HHT-α method



dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]

vn+1 = vn + ∆t [(1 − γ) an + γan+1]

(5)

where:

dn+1−α = (1 + α) dn+1 − αdn

vn+1−α = (1 + α) vn+1 − αvn

tn+1−α = (1 + α) tn+1 − αtn(6)

If α = 0, the HHT-α method is reduced to Newmark’s method.When applied to d ′′ + ω2d = 0, the method takes the recursive form Xn+1 = AXn, where A isthe amplification matrix.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

HHT-α method



dn+1 = dn + ∆tvn + ∆t22 [(1 − 2β) an + 2βan+1]

vn+1 = vn + ∆t [(1 − γ) an + γan+1]

(5)

where:

dn+1−α = (1 + α) dn+1 − αdn

vn+1−α = (1 + α) vn+1 − αvn

tn+1−α = (1 + α) tn+1 − αtn(6)

If α = 0, the HHT-α method is reduced to Newmark’s method.When applied to d ′′ + ω2d = 0, the method takes the recursive form Xn+1 = AXn, where A isthe amplification matrix.Similarly to Newmark, HHT-α method can also be reduced to a three-step linear multistepmethod for second order differential equations:

3∑

i=0

αi dn+i = h23∑

i=0

βi d′′

n+i (7)


α0 = 0, β0 = γα − 12 α − βα

α1 = 1, β1 = −2γα + 3βα − γ + β + 12

α2 = −2, β2 = β(−3α − 2) +(γ + 1

2

)(1 + α)

α3 = 1, β3 = β + βα

(8)


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

HHT-α method

The method is second-order accurate when γ = 1−2α2 .


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

HHT-α 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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

HHT-α 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 ωh/(2π) = Ω/(2π).


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Numerical methods for first order ODEs

Consider a first order ODE: y ′(t) = f (t, y(t)), y(a) = y0

Linear multistep methods ⇒ BDFs ⇒ ode15s


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Numerical methods for first order ODEs

Consider a first order ODE: y ′(t) = f (t, y(t)), y(a) = y0

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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

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).


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Linear multistep methods

Linear multistep methods:k∑

j=0

αj yn+j = hk∑

j=0

βj fn+j


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



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

The error estimation that the ode15s uses is the local truncation error which results large in vibratingproblems:

est ≈ LTE = Chk+1yk+1(tn) + O

(

hk+2)

(9)

−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.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



(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.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



(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


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

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.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



BDF2: 32 yn+2 − 2yn+1 + 1

2 yn = hfn+2


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)


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions



BDF2: 32 yn+2 − 2yn+1 + 1

2 yn = hfn+2


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


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Introduction


First orderODEs

BDF-αmethod

Results

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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


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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

)

−(

∑ 2i=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


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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

)

−(

∑ 2i=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


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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

)

−(

∑ 2i=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



Second order BDF-α:(

3

2+ α

)

yn+2 + (−2 − 2α) yn+1 +

(1

2+ α

)

yn = h(1 + α)fn+2 − hαfn+1


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Introduction


First orderODEs

BDF-αmethod

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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

)

−(

∑ 2i=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



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


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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


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Stability regions


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


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Introduction


First orderODEs

BDF-αmethod

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Stability regions


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

]


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First orderODEs

BDF-αmethod

Results

Conclusions

Stability regions


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


[(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, +∞)


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Introduction


First orderODEs

BDF-αmethod

Results

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Stability regions


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


[(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


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BDF-αmethod

Results

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Studying the dissipation → analysis of the amplification factorThe 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

)


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions


Studying the dissipation → analysis of the amplification factorThe 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)(10)

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].


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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


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Results

Conclusions

Results

Example 1: Linear wave equation with rectangular pulse IC (1D).400 elements and 1400 steps.


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Results


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1


α=0.3 γ=0.8 β=0.4225

t= 0t= 2


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Results


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1


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


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Results


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1


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


α=−0.35

Computational times → HHT-α: 1.88 seconds, BDF-α: 6.17 secondsBoth very quick but HHT-α 3 times quicker.


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First orderODEs

BDF-αmethod

Results

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Results


0 1 2 3 4 5 6 7 8−0.5

0

0.5

1


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


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 .


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First orderODEs

BDF-αmethod

Results

Conclusions

Results

Example 2: Non-linear PDE of a guitar string.20 elements and 1000 steps.


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BDF-αmethod

Results

Conclusions

Results


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


α=0.3 , γ=0.8, β=0.4225

Figure: HHT-α = 0.3.


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Results


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


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

0.08

α=−0.35

Método BDF−α, tiempo=0.015198, nele=20, pasos=1000, masa=cons

Figure: BDF-α = −0.35.


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First orderODEs

BDF-αmethod

Results

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Results


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


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

0.08

α=−0.35



Computational times → HHT-α = 0.3: 244.76 seconds, 9479 iterations. BDF-α = −0.35:259.89 seconds, 9516 iterations.


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First orderODEs

BDF-αmethod

Results

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Results


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


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

0.08

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: 0.1 seconds and BDF-α1.54 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.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Conclusions

A new method called BDF-α has been built.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

Conclusions


Controlled numerical dissipation in the medium and high-frequency range whenapplied to second order ODEs modelling vibratory systems is a desirable propertywhen dealing with second order ODE systems associated to the FEMsemidiscretization of the wave-type PDEs.


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BDF-αmethod

Results

Conclusions

Conclusions



The BDF-α method is a parametrized second-order accurate multistep method,A-stable when α ∈ [−0.5, +∞) and which allows controlled numerical dissipation inthe medium and high-frequency range when applied to second order ODEs modellingvibratory systems.


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First orderODEs

BDF-αmethod

Results

Conclusions

Conclusions



The BDF-α method is a parametrized second-order accurate multistep method,A-stable when α ∈ [−0.5, +∞) and which allows controlled numerical dissipation inthe medium and high-frequency range when applied to second order ODEs modellingvibratory systems.

The BDF-α method improves the constant error of the BDF2 and its spectral radii ρ∞

sweeps the whole interval [0, 1] offering similar numerical damping control as theHHT-α method.


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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

F. Armero, I. Romero, On the formulation of high-frequency dissipative time stepping algorithms for

nonlinear dynamics. Part I: low order methods for two model problems and nonlinear elastodynamics,Comput. Meth. Appl. Mech. Eng., 190 (2000), 2603-2649.

F. Armero, I. Romero, On the formulation of high-frequency dissipative time stepping algorithms for

nonlinear dynamics. Part II: second order methods, Comput. Meth. Appl. Mech. Eng., 190 (2001),6783-6824.

J.C. Butcher, Numerical methods for ordinary differential equations in the 20th century, J. Comput.

Appl. Math., 125 (2000), 1-29.

J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley& Sons, Chichester,

2008.

J.R. Cash, On the integration of stiff systems of ODEs using extended backward differentiation

formulae, Numer. Math., 34:2 (1980), 235-246.

J.R. Cash, Second derivative extended backward differentation formulas for the numerical integration

of stiff systems, SIAM J. Numer. Anal., 18:1 (1981), 21-36.

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differentiation formula, Comput. Math. Appl., 9:5 (1983), 645-657.

J.R. Cash, S. Considine, An MEBDF code for stiff initial value problems, ACM Trans. Math. Software,

18:2 (1992), 142-155.

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dissipation: the generalized-α method, Journal of Applied Mechanics, 60, 371-375, (1993).

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Introduction


First orderODEs

BDF-αmethod

Results

Conclusions

W.H. Enright, Second derivative multistep methods for stiff ordinary differential equations, SIAM J.

Numer. Anal., 11:2 (1974), 321-331.

C. Fredebeul, A-BDF: a generalization of the backward differentiation formulae, SIAM J. Numer. Anal.,

35:5 (1998), 1917-1938.

C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, New

Jersey, 1971.

I. Gladwell, R. Thomas, Stability properties of the Newmark, Houbolt and Wilson θ methods, Int. J.

Numer. Anal. Methods Geomech., 4 (1980), 143-158.

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elasticity, Comput. Methods Appl. Mech. Eng., 190 (2000), 1763-1783.

E. Hairer, S.P. Nørsett, G. Wanner, Solving ordinary differential equations, I, Nonstiff problems,

Springer, Berlin, 1993.

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Problems, Springer, Berlin, 1996.

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structural dynamics, Earthq. Eng. Struct. Dyn., 6 (1978), 99-117.

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algorithms in structural dynamics, Earthq. Eng. Struct. Dyn., 5 (1977), 283-292.

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systems of ODEs, Math. Comput. Simul., 66 (2004), 33-41.


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First orderODEs

BDF-αmethod

Results

Conclusions

G. Hojjati, M.Y.R. Ardabili, S.M. Hosseini, New second derivative multistep methods for stiff systems,

Appl. Math. Model., 30 (2006), 466-476.

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