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Fourth-order energy-preserving locally implicit time discretization for linear wave equations
Juliette Chabassier, Sebastien Imperiale
To cite this version:
Juliette Chabassier, Sebastien Imperiale. Fourth-order energy-preserving locally implicit time dis- cretization for linear wave equations. WAVES 2015 - The 12th international conference on mathemat- ical and numerical aspects of wave propagation, Jul 2015, Karlsruhe, Germany. �10.1002/nme.5130�.
�hal-01264044�
Fourth order energy-preserving locally implicit discretization for linear wave equations
Juliette Chabassier 1,∗ , S´ ebastien Imperiale 2
1 Magique3D Team, Inria Bordeaux Sud Ouest, University of Pau and Pays de l’Adour, Pau, France.
2 M3DISIM Team, Inria Saclay, Palaiseau, France
∗ Email: [email protected] Suggested Scientific Committee Members:
Eliane B´ ecache, Julien Diaz, Jer´ onimo Rodr´ıguez Garc´ıa Abstract
A family of fourth order locally implicit schemes is presented as a special case of fourth order cou- pled implicit schemes for linear wave equations.
The domain of interest is decomposed into sev- eral regions where different (explicit or implicit) fourth order time discretization are used. The coupling is based on a Lagrangian formulation on the boundaries between the several non con- forming meshes of the regions. Fourth order ac- curacy follows from global energy identities. Nu- merical results in 1d and 2d illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using everywhere an explicit scheme can be ex- tremely penalizing. Fourth order accuracy re- duces the numerical dispersion inherent to im- plicit methods used with a large time step, and makes this family of schemes attractive com- pared to classical approaches.
Keywords: High-order numerical methods, Time discretization, Locally implicit schemes.
1 Continuous system
We want to solve for time t > 0, the system (closed with Neumann homogeneous boundary conditions):
∂ t 2 u 0 − ∇ · c 2 (x) ∇u 0 = s 0 in Ω 0 , (1a) c 2 (x) ∇u 0 · n 0 = λ on Γ, (1b)
∂ t 2 u 1 − ∇ · c 2 (x) ∇u 1 = s 1 in Ω 1 , (1c) c 2 (x) ∇u 1 · n 1 = −λ on Γ, (1d)
u 0 = u 1 on Γ (1e)
in a domain Ω composed by disjoint sets Ω = Ω 0 ∪ Ω 1 separated by Γ = Ω 0 ∩ Ω 1 . s 0 and s 1
are given source terms, and c(x) > c 0 > 0 is the inhomogeneous velocity of the waves. Any solution to (1) satisfies the energy identity:
dE 01 dt =
Z
Ω
0s 0 ∂ t u 0 + Z
Ω
1s 1 ∂ t u 1 , where E 01 = 1
2 k∂ t u 0 k 2 L
2(Ω
0) + 1
2 k∂ t u 1 k 2 L
2(Ω
1)
+ 1
2 kc ∇u 0 k 2 L
2(Ω
0) + 1
2 kc ∇u 1 k 2 L
2(Ω
1) (2) 2 Semi discrete system
We consider spatial meshes of Ω 0 and Ω 1 upon which are based finite dimensional finite element spaces: V h,0 ⊂ H 1 (Ω 0 ), V h,1 ⊂ H 1 (Ω 1 ) and Γ h ⊂ H −1/2 (Γ). One have leeway in the choice of (V h,0 , V h,1 ) after which Γ h must be chosen so that an inf-sup type condition is satisfied, see [1, 3, 4]. ( U e h,0 , U e h,1 , Λ e h ) is the solution of:
d 2 t M h,0 U e h,0 + K h,0 U e h,0 − t C h,0 Λ e h = M h,0 S e h,0 (3a) d 2 t M h,1 U e h,1 + K h,1 U e h,1 + t C h,1 Λ e h = M h,1 S e h,1 (3b) C h,0 U e h,0 = C h,1 U e h,1 (3c) A semi discrete energy identity can be obtained.
dE 01,h
dt = M h,0 S e h,0 ·d t U e h,0 +M h,1 S e h,1 ·d t U e h,1 , where E 01,h = 1
2 kd t U e h,0 k 2 M
h,0
+ 1
2 kd t U e h,1 k 2 M
h,1
+ 1
2 k U e h,0 k 2 K
h,0
+ 1
2 k U e h,1 k 2 K
h,1
(4) where kXk 2 M = M X · X for any nonnegative matrix M .
3 Discrete system
The proposed numerical discretization is based on the following definitions:
D ∆t 2 U h n := U h n+1 − 2U h n + U h n−1
/∆t 2
{U h } n θ := θ U h n+1 + (1 − 2θ)U h n + θ U h n−1
1
The consistency analysis of the fourth order fam- ily of schemes [2] applied to each equation of system (3) instigates the following scheme:
M h,0 D 2 ∆t U h,0 n + K h,0 {U h,0 } n θ
0
− t C h,0 Π n h = M h,0 S h,0 n +∆t 2 α 0 K h,0 M h,0 −1
−K h,0 {U h,0 } n ϕ
0
+ t C h,0 Π n h (5a) M h,1 D 2 ∆t U h,1 n + K h,1 {U h,1 } n θ
1
+ t C h,1 Π n h = M h,1 S h,1 n +∆t 2 α 1 K h,1 M h,1 −1
−K h,1 {U h,1 } n ϕ
1
− t C h,1 Π n h (5b) C h,0 U h,0 n+1 − U h,0 n−1
2∆t − C h,1 U h,1 n+1 − U h,1 n−1
2∆t = 0 (5c)
where α i = θ i − 1/12. Any solution to (5) satis- fies the energy identity:
E 01,4,h n+1/2 − E 01,4,h n+1/2
∆t = I e h,0 −1 M h,0 S h,0 n · U h,0 n+1 − U h,0 n−1 2∆t + I e h,1 −1 M h,1 S h,1 n · U h,1 n+1 − U h,1 n−1
2∆t ,
where the discrete energy reads E 01,4,h n+1/2 = 1
2
U h,0 n+1 − U h,0 n
∆t
2
M c
h,0+ 1 2
U h,1 n+1 − U h,1 n
∆t
2
M c
h,1+ 1 2
U h,0 n+1 + U h,0 n 2
2
K
h,0+ 1 2
U h,1 n+1 + U h,1 n 2
2
K
h,1where the modified mass matrices M c h,i are de- fined by M c h,i = I e h,i −1 M f h,i where
I e h,i = I h,i + ∆t 2
θ i − 1 12
K h,i M h,i −1
M f h,i = M h,i + ∆t 2
θ i − 1 4
K h,i +∆t 4
θ i − 1
12 ϕ i − 1 4
K h,i M h,i −1 K h,i The positivity of the energy can be proven un- der standard CFL condition that depend on the parameters (θ i , ϕ i ). Despite the non standard form of this energy, stability in L2-norm can be proved in the case θ i ≥ 1/4 and ϕ i ≥ 1/4. L2- Stability in the other cases is not proven yet but show good numerical behavior.
A 1d numerical experiment is performed where the segment [0, 1] is cut in two intervals Ω 0 = [0, 0.5] and Ω 1 = [0.5, 1]. Ω 0 and Ω 1 are respec- tively divided into 7 and 13 elements. Sixth or- der spectral elements are implemented. A fourth
order explicit scheme is used on Ω 0 (θ 0 = ϕ 0 = 0) while an unconditionally stable implicit scheme is used on Ω 1 (θ 1 = ϕ 1 = 1/4). A gaussian initial condition is set on the left interval and crosses the middle point around time 0.3. Fig. 1 shows that the energy is preserved up to machine precision. Fig. 2 shows that the coupling of sec- ond order implicit and explicit schemes only pro- vides second order accuracy (as expected), while our scheme provides fourth order accuracy.
Figure 1: Relative energy deviation
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−15
−10
−5 0
x 10−15
time (s)
relative energy deviation
Figure 2: Convergence curve
10−3 10−7
10−6 10−5 10−4 10−3 10−2
Time Step
Relative Error
2
4
0 (Explicit) − 1/4 (Implicit) (0−0) (Explicit) − (1/4,1/4) (Implicit)
