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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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave equation
PDEs→ FEM approximation
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
Wave:
utt = α2uxx
CC : u(0, t) = 0 = u(L, t)
CI : u(x , 0) = g(x), ut(x , 0) = 0
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
Wave:
utt = α2uxx
CC : u(0, t) = 0 = u(L, t)
CI : u(x , 0) = g(x), ut(x , 0) = 0
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
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)
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
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)
Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
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)
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
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)
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
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)
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
A FEM application to the 1D linear diffusionand wave 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]
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)
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
CONTINUOUS
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
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.
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
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.
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
400 elements:
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear wave equation examples in MATLAB
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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:
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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:
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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:
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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:
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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:
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
- 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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
- 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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
- 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.
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)
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
- 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.
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)
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
- 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.
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)
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear multistep methods for 2nd order ODEs
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)
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear multistep methods for 2nd order ODEs
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)
Applying the order conditions for linear multistep methods, the method results second-orderaccurate when γ = 1/2.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear multistep methods for 2nd order ODEs
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)
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.When applied to d ′′ + ω2d = 0, the method takes the recursive form Xn+1 = AXn, where A isthe amplification matrix.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.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)
where the coefficients αj , βj are given by:
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)
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
HHT-α method
The method is second-order accurate when γ = 1−2α2 .
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
HHT-α method
The method is second-order accurate when γ = 1−2α2 .
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
HHT-α method
The method is second-order accurate when γ = 1−2α2 .
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).
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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).
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Linear multistep methods
Linear multistep methods:k∑
j=0
αj yn+j = hk∑
j=0
βj fn+j
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
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.
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
)
−(
∑ 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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
)
−(
∑ 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
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
)
−(
∑ 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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
)
−(
∑ 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
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Considerations about the new method
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
)
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Considerations about the new method
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].
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Results
Example 1: Linear wave equation with rectangular pulse IC (1D).400 elements and 1400 steps.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Results
Example 2: Non-linear PDE of a guitar string.20 elements and 1000 steps.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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: 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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Conclusions
A new method called BDF-α has been built.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Conclusions
A new method called BDF-α has been built.
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Conclusions
A new method called BDF-α has been built.
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.
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Conclusions
A new method called BDF-α has been built.
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.
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.
Numericalmethods forstiff ODEs
ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
F. Armero, I. Romero, On the formulation of high-frequency dissipative time stepping algorithms for
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ElisabeteAlberdi
Celaya1 , JuanJose Anza2
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LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
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ElisabeteAlberdi
Celaya1 , JuanJose Anza2
Introduction
LMS forsecond orderODEs
First orderODEs
BDF-αmethod
Results
Conclusions
Thank You foryour attention