efficient space and time discretization of the wave equation ......e cient space and time...
TRANSCRIPT
![Page 1: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/1.jpg)
Efficient Space and Time Discretization of theWave Equation. Application to the Reverse Time
Migration
J. Diaz1,2
In collaboration with C. Baldassari1,2, H. Barucq1,2,H. Calandra3, B. Denel3
(1) INRIA Bordeaux Research Center, Project Team Magique3D(2) LMA, CNRS UMR 5142, Universite de Pau(3) TOTAL
08/06/2010
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Outline
1 What is the Reverse Time Migration?
2 Appropriate space discretization of the wave equation
3 Improvement of the time discretization of the wave equation
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The seismic reflection
Oil exploration by seismic reflection :
Why? Have a good image of the sub-surface andconsequently know where there is oil or not.
How? Seismic acquisition campaigns.
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Seismic acquisition : principle
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Seismic Imaging (summary)
1 Get an initial velocity model from acquisition campain:
Tomography step
2 Using same sources and receivers as for tomography
Solve the waveequation twice foreach source
(a) Propagate the sources into the velocitymodel
(b) Retropropagate the recorded waves(reverse time)
3 Cross-correlation of (a) and (b): imaging condition(Claerbout)
Produce an image of ground
4 Compare with velocity model: if different, go to 2 aftermodification of the velocity model.
![Page 6: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/6.jpg)
Seismic Imaging (summary)
1 Get an initial velocity model from acquisition campain:
Tomography step
2 Using same sources and receivers as for tomography
Solve the waveequation twice foreach source
(a) Propagate the sources into the velocitymodel
(b) Retropropagate the recorded waves(reverse time)
3 Cross-correlation of (a) and (b): imaging condition(Claerbout)
Produce an image of ground
4 Compare with velocity model: if different, go to 2 aftermodification of the velocity model.
![Page 7: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/7.jpg)
Seismic Imaging (example)
Seismogram from theacquisition campaign
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Seismic Imaging (example)
Seismogram from theacquisition campaign
=⇒
Initial guess
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Seismic Imaging (example)
Computational Domain
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Seismic Imaging (example)
Computational Domain
=⇒
RTM Image
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Numerical methods for solving the wave equation
∂2u
∂t2− c2∆u = f (t, x), (t, x) ∈ [0 ; T ]× Ω
• Finite difference methods: fast but not really effective inheterogeneous medium.
• Finite element methods with the choice criteria:
• Efficient to take the topography effects into account;
• Optimized computational burden;
• Limitation of the dispersion effects.
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Numerical methods for solving the wave equation
M∂2U
∂t2+ KU = F
• Finite difference methods: fast but not really effective inheterogeneous medium.
• Finite element methods with the choice criteria:
• Efficient to take the topography effects into account;
• Optimized computational burden;
• Limitation of the dispersion effects.
![Page 13: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/13.jpg)
Numerical methods for solving the wave equation
Most popular Finite Element Method (FEM) for geophysicalapplications:
Spectral Element Method (SEM) (Seriani, Priolo, Cohen, Komatitsch...)
• Explicit scheme (diagonal mass matrix)
• In general, FEM =⇒ implicit scheme;
• Combined with a lumping process: the order of convergence isdestroyed except for order one or SEM.
![Page 14: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/14.jpg)
Numerical methods for solving the full-wave equation
Discontinuous Galerkin Method (DG)
• Quasi-explicit scheme: mass matrix is block-diagonal;
• Based on tetrahedras (easy to handle);
• Adapted to strongly varying velocities;
• Fast computational methods:
• Volume calculi are made locally;
• Communications between elements thanks to conditions on theedges only ((N − 1)D calculi);
• High-order elements to overcome dispersion effects.
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SEM versus DG
1 The SEM method• Gauss Lobatto quadrature rule : diagonal mass matrix, do not
hamper the order of convergence.• Meshes made of quadrangles in 2D or hexahedra in 3D :
difficult to compute, not always suitable to complextopographies.
2 The DG method• Representation of the solution is quasi-explicit because the
mass matrix is block-diagonal without any approximation.• To compute easily its coefficients, we use an exact quadrature
formula which does not hamper the order of convergence.• Meshes made of triangles in 2D or tetrahedra in 3D. Thus the
topography of the computational domain is easily discretized.• Handle polynomial velocities inside each element.
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Interior Penalty Discontinuous Galerkin Method (IPDG)
• V hl :=
v ∈ L2 (Ω) : v|K ∈ Pl (K ) ∀K ∈ Th
• M is block-diagonal
• Kij =∑K∈Th
∫K
1
ρ∇v i · ∇v j dx −
∑F∈F
∫F
[[ v j ]] · 1
ρ∇v i dF
-∑F∈F
∫F
[[ v i ]] · 1
ρ∇v j dF +
∑F∈F
∫F
γ[[ v i ]] · [[ v j ]] dF
(Ainswoth et al., Grote et al.)
![Page 17: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/17.jpg)
Interior Penalty Discontinuous Galerkin Method (IPDG)
• V hl :=
v ∈ L2 (Ω) : v|K ∈ Pl (K ) ∀K ∈ Th
• M is block-diagonal
• Kij =∑K∈Th
∫K
1
ρ∇v i · ∇v j dx −
∑F∈F
∫F
[[ v j ]] · 1
ρ∇v i dF
-∑F∈F
∫F
[[ v i ]] · 1
ρ∇v j dF
+∑F∈F
∫F
γ[[ v i ]] · [[ v j ]] dF
(Ainswoth et al., Grote et al.)
![Page 18: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/18.jpg)
Interior Penalty Discontinuous Galerkin Method (IPDG)
• V hl :=
v ∈ L2 (Ω) : v|K ∈ Pl (K ) ∀K ∈ Th
• M is block-diagonal
• Kij =∑K∈Th
∫K
1
ρ∇v i · ∇v j dx −
∑F∈F
∫F
[[ v j ]] · 1
ρ∇v i dF
-∑F∈F
∫F
[[ v i ]] · 1
ρ∇v j dF +
∑F∈F
∫F
γ[[ v i ]] · [[ v j ]] dF
(Ainswoth et al., Grote et al.)
![Page 19: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/19.jpg)
Interior Penalty Discontinuous Galerkin Method (IPDG)
• V hl :=
v ∈ L2 (Ω) : v|K ∈ Pl (K ) ∀K ∈ Th
• M is block-diagonal
• Kij =∑K∈Th
∫K
1
ρ∇v i · ∇v j dx −
∑F∈F
∫F
[[ v j ]] · 1
ρ∇v i dF
-∑F∈F
∫F
[[ v i ]] · 1
ρ∇v j dF +
∑F∈F
∫F
γ[[ v i ]] · [[ v j ]] dF
(Ainswoth et al., Grote et al.)
![Page 20: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/20.jpg)
“Drawbacks” of the DG methods
• The solution to the wave equation is not discontinuous.
The solution is neither piecewise polynomial. Actually, thediscontinuities of the approximate solution are controlled bythe penalization parameter and they do not hamper theaccuracy of the method.
• DG requires more degrees of freedom than SEM.
That is true for a given mesh. But there is no reason to usethe same mesh for both methods.
• The stability condition (which restricts the time space of thescheme) is smaller for DG than for SEM.
That is true for a given mesh.
![Page 21: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/21.jpg)
“Drawbacks” of the DG methods
• The solution to the wave equation is not discontinuous.
The solution is neither piecewise polynomial. Actually, thediscontinuities of the approximate solution are controlled bythe penalization parameter and they do not hamper theaccuracy of the method.
• DG requires more degrees of freedom than SEM.
That is true for a given mesh. But there is no reason to usethe same mesh for both methods.
• The stability condition (which restricts the time space of thescheme) is smaller for DG than for SEM.
That is true for a given mesh.
![Page 22: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/22.jpg)
“Drawbacks” of the DG methods
• The solution to the wave equation is not discontinuous.
The solution is neither piecewise polynomial. Actually, thediscontinuities of the approximate solution are controlled bythe penalization parameter and they do not hamper theaccuracy of the method.
• DG requires more degrees of freedom than SEM.
That is true for a given mesh. But there is no reason to usethe same mesh for both methods.
• The stability condition (which restricts the time space of thescheme) is smaller for DG than for SEM.
That is true for a given mesh.
![Page 23: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/23.jpg)
“Drawbacks” of the DG methods
• The solution to the wave equation is not discontinuous.
The solution is neither piecewise polynomial. Actually, thediscontinuities of the approximate solution are controlled bythe penalization parameter and they do not hamper theaccuracy of the method.
• DG requires more degrees of freedom than SEM.
That is true for a given mesh. But there is no reason to usethe same mesh for both methods.
• The stability condition (which restricts the time space of thescheme) is smaller for DG than for SEM.
That is true for a given mesh.
![Page 24: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/24.jpg)
“Drawbacks” of the DG methods
• The solution to the wave equation is not discontinuous.
The solution is neither piecewise polynomial. Actually, thediscontinuities of the approximate solution are controlled bythe penalization parameter and they do not hamper theaccuracy of the method.
• DG requires more degrees of freedom than SEM.
That is true for a given mesh. But there is no reason to usethe same mesh for both methods.
• The stability condition (which restricts the time space of thescheme) is smaller for DG than for SEM.
That is true for a given mesh.
![Page 25: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/25.jpg)
“Drawbacks” of the DG methods
• The solution to the wave equation is not discontinuous.
The solution is neither piecewise polynomial. Actually, thediscontinuities of the approximate solution are controlled bythe penalization parameter and they do not hamper theaccuracy of the method.
• DG requires more degrees of freedom than SEM.
That is true for a given mesh. But there is no reason to usethe same mesh for both methods.
• The stability condition (which restricts the time space of thescheme) is smaller for DG than for SEM.
That is true for a given mesh.
![Page 26: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/26.jpg)
What are the good criteria to compare IPDG and SEM?
• Compare the computational burden for a given accuracy.
• Compare the accuracy for a given computational burden.
The computation cost of one iteration is directly related tothe number of degrees of freedom.=⇒ compare the accuracy for a given number of degrees of
freedom
![Page 27: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/27.jpg)
What are the good criteria to compare IPDG and SEM?
• Compare the computational burden for a given accuracy.
• Compare the accuracy for a given computational burden.
The computation cost of one iteration is directly related tothe number of degrees of freedom.=⇒ compare the accuracy for a given number of degrees of
freedom
![Page 28: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/28.jpg)
What are the good criteria to compare IPDG and SEM?
• Compare the computational burden for a given accuracy.
• Compare the accuracy for a given computational burden.The computation cost of one iteration is directly related tothe number of degrees of freedom.
=⇒ compare the accuracy for a given number of degrees offreedom
![Page 29: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/29.jpg)
What are the good criteria to compare IPDG and SEM?
• Compare the computational burden for a given accuracy.
• Compare the accuracy for a given computational burden.The computation cost of one iteration is directly related tothe number of degrees of freedom.=⇒ compare the accuracy for a given number of degrees of
freedom
![Page 30: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/30.jpg)
Comparison IPDG versus SEM
To obtain a given accuracy:
• The number of degrees of freedom and the number ofmultiplications required by IPDG are (slightly) smaller.
• The stability condition of IPDG is higher (and the number ofiterations is smaller).
IPDG performs as well as (and sometimes better than) SEM.
![Page 31: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/31.jpg)
A more realistic experiment
![Page 32: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/32.jpg)
A more realistic experiment
![Page 33: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/33.jpg)
Improvements of the method
• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh (p-adaptivity)
![Page 34: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/34.jpg)
Improvements of the method
• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh (p-adaptivity)
![Page 35: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/35.jpg)
Improvements of the method
![Page 36: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/36.jpg)
Improvements of the method
![Page 37: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/37.jpg)
Improvements of the method
• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh
• Use a local time stepping strategy
• Use a second order time scheme in the fine mesh and a fourthorder time scheme in the coarse mesh.
![Page 38: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/38.jpg)
Improvements of the method
• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh
• Use a local time stepping strategy
• Use a second order time scheme in the fine mesh and a fourthorder time scheme in the coarse mesh.
![Page 39: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/39.jpg)
Improvements of the method
• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh
• Use a local time stepping strategy
• Use a second order time scheme in the fine mesh and a fourthorder time scheme in the coarse mesh.
![Page 40: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/40.jpg)
Improvements of the method
• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh
• Use a local time stepping strategy
• Use a second order time scheme in the fine mesh and a fourthorder time scheme in the coarse mesh.
![Page 41: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/41.jpg)
Local Time Stepping : Bibliography
POems Team: Becache, Collino, Fouquet, Joly, Rodrıguez
• Conservation of energy
• Optimal stability condition
• Requires the introduction of a Lagrange Multiplier
• Implicit scheme on the interface
Piperno
• First-order Maxwell system
• Conservation of energy
• Optimal stability condition
• Explicit or implicit scheme on the interface
![Page 42: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/42.jpg)
Local Time Stepping : Bibliography
POems Team: Becache, Collino, Fouquet, Joly, Rodrıguez
• Conservation of energy
• Optimal stability condition
• Requires the introduction of a Lagrange Multiplier
• Implicit scheme on the interface
Piperno
• First-order Maxwell system
• Conservation of energy
• Optimal stability condition
• Explicit or implicit scheme on the interface
![Page 43: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/43.jpg)
Local Time Stepping : Bibliography
ADER Schemes, Kaser, Dumbser et al.
• High-order Explicit Time Schemes ;
• First order Systems ;
• No energy conservation ;
• Difficult to implement (one time step by element).
![Page 44: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/44.jpg)
Local Time Stepping : Bibliography
Hairer, Lubich and Wanner (2002), Leimkuhler and Reich (2004) :Local Time Stepping for ODE’s (second order scheme)
Diaz, Grote (2009) : High-Order Local Time Stepping for theWave Equation.
• Conservation of energy
• Optimal stability condition
![Page 45: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/45.jpg)
Local Time Stepping : Bibliography
Hairer, Lubich and Wanner (2002), Leimkuhler and Reich (2004) :Local Time Stepping for ODE’s (second order scheme)
Diaz, Grote (2009) : High-Order Local Time Stepping for theWave Equation.
• Conservation of energy
• Optimal stability condition
![Page 46: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/46.jpg)
Local Time Stepping
![Page 47: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/47.jpg)
Local Time Stepping
![Page 48: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/48.jpg)
Conclusions
• The performances of IPDG and SEM are similar with highorder polynomials in 2D.
• It is much easier to use elements of various order with IPDG.
• The local time stepping strategy does not hamper theaccuracy of the method.
• Comparison of IPDG and SEM in 3D.
• Extension to elastodynamics.
![Page 49: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/49.jpg)
Time discretization of the wave equation
Md2U
dt2+ KU = 0
We consider space discretization methods such that M and K aresymmetric positive matrices and M is (block-)diagonal (FEM withmass lumping or DG methods).
Classical Leap Frog Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= + O(∆t2)
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 50: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/50.jpg)
Time discretization of the wave equation
M12d2U
dt2+ M−
12KM−
12︸ ︷︷ ︸
A
M12U︸ ︷︷ ︸Y
= 0
Classical Leap Frog Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= + O(∆t2)
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 51: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/51.jpg)
Time discretization of the wave equation
d2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= + O(∆t2)
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 52: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/52.jpg)
Time discretization of the wave equation
d2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= + O(∆t2)
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 53: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/53.jpg)
Time discretization of the wave equation
d2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2=
d2Y
dt2(t) + O(∆t2)
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 54: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/54.jpg)
Time discretization of the wave equation
d2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2=
d2Y
dt2(t) + O(∆t2)
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 55: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/55.jpg)
Time discretization of the wave equation
d2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= AY (t) + O(∆t2)
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 56: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/56.jpg)
Time discretization of the wave equation
d2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y n+1 − 2Y n + Y n−1
∆t2= −AY n.
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 57: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/57.jpg)
Time discretization of the wave equation
d2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y n+1 − 2Y n + Y n−1
∆t2= −AY n.
Energy Conservation
En+ 12 =
⟨Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩+⟨AY n+1,Y n
⟩
![Page 58: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/58.jpg)
Time discretization of the wave equationd2Y
dt2+ AY = 0
Classical Leap Frog Scheme:
Y n+1 − 2Y n + Y n−1
∆t2= −AY n.
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4A
)Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨AY n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 59: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/59.jpg)
Time discretization of the wave equation
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4A
)Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨AY n+1 + Y n
2,Y n+1 + Y n
2
⟩CFL Condition
The scheme is stable, if and only if :
I − ∆t2
4A and A are symmetric positive
![Page 60: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/60.jpg)
Time discretization of the wave equation
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4A
)Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨AY n+1 + Y n
2,Y n+1 + Y n
2
⟩CFL Condition
The scheme is stable, if and only if :
I − ∆t2
4A and A are symmetric positive
0 ≤ λA ≤4
∆t2
![Page 61: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/61.jpg)
Time discretization of the wave equation
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4A
)Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨AY n+1 + Y n
2,Y n+1 + Y n
2
⟩CFL Condition
The scheme is stable under the CFL condition :
∆t ≤ αLFh
![Page 62: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/62.jpg)
Time discretization of the wave equation
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4A
)Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨AY n+1 + Y n
2,Y n+1 + Y n
2
⟩CFL Condition
The scheme is stable under the CFL condition :
∆t ≤ αLFh
![Page 63: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/63.jpg)
Time discretization of the wave equation
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4A
)Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨AY n+1 + Y n
2,Y n+1 + Y n
2
⟩CFL Condition
We want the new scheme to satisfy:
∆tcoarse ≤ αLFhcoarse and ∆tfine ≤ αLFh
fine
≈ αLFhcoarse/p
![Page 64: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/64.jpg)
Time discretization of the wave equation
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4A
)Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨AY n+1 + Y n
2,Y n+1 + Y n
2
⟩CFL Condition
We want the new scheme to satisfy:
∆tcoarse ≤ αLFhcoarse and ∆tfine ≤ αLFh
fine ≈ αLFhcoarse/p
![Page 65: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/65.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:.
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 66: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/66.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2=
d2Y
dt2(t) +
∆t2
12
d4Y
dt4(t) + O(∆t4).
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 67: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/67.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2=
d2Y
dt2(t) +
∆t2
12
d4Y
dt4(t) + O(∆t4).
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 68: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/68.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= AY (t) +
∆t2
12Ad2Y
dt2(t) + O(∆t4).
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 69: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/69.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= AY (t) +
∆t2
12Ad2Y
dt2(t) + O(∆t4).
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 70: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/70.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:
Y (t + ∆t)− 2Y (t) + Y (t −∆t)
∆t2= AY (t) +
∆t2
12A2Y (t) + O(∆t4).
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 71: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/71.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:
Y n+1 − 2Y n + Y n−1
∆t2= −AY n +
∆t2
12A2Y n.
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 72: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/72.jpg)
Higher Order Schemes, Global Time Stepping
d2Y
dt2+ AY = 0
Modified Equation Scheme:
Y n+1 − 2Y n + Y n−1
∆t2= −AY n +
∆t2
12A2Y n.
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
![Page 73: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/73.jpg)
Higher Order Schemes, Global Time Stepping
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
CFL Condition
The scheme is stable, if and only if
I − ∆t2
4
(A− ∆t2
12A2
)and A− ∆t2
12A2 are symmetric positive
![Page 74: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/74.jpg)
Higher Order Schemes, Global Time Stepping
Energy Conservation
En+ 12 =
⟨(I − ∆t2
4
(A− ∆t2
12A2
))Y n+1 − Y n
∆t,Y n+1 − Y n
∆t
⟩
+
⟨(A− ∆t2
12A2
)Y n+1 + Y n
2,Y n+1 + Y n
2
⟩
CFL Condition
The scheme is stable under the CFL condition
∆t ≤ αMEh =√
3αLFh
![Page 75: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/75.jpg)
Higher Order Schemes, Global Time Stepping
CFL Condition
The scheme is stable under the CFL condition
∆t ≤ αMEh =√
3αLFh
![Page 76: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/76.jpg)
Higher Order Schemes, Global Time Stepping
CFL Condition
We want the new scheme to satisfy
∆tcoarse ≤ αMEhcoarse and ∆tfine ≤ αMEh
fine
≈ αMEhcoarse/p
![Page 77: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/77.jpg)
Higher Order Schemes, Global Time Stepping
CFL Condition
We want the new scheme to satisfy
∆tcoarse ≤ αMEhcoarse and ∆tfine ≤ αMEh
fine ≈ αMEhcoarse/p
![Page 78: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/78.jpg)
Auxiliary Function
At each time step n we define an auxiliary function
Qn(τ) =Y (n∆t − τ) + Y (n∆t + τ)
2
for τ ∈ [−∆t ; ∆t].
This function is obviously even and satisfy:d2Qn
dτ2(τ) = −AQn(τ),
Qn(0) = Y (n∆t),dQn
dτ(0) = 0,
After having solved this equation, Y ((n + 1)∆t) can be computedusing Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t)
![Page 79: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/79.jpg)
Auxiliary Function
At each time step n we define an auxiliary function
Qn(τ) =Y (n∆t − τ) + Y (n∆t + τ)
2
for τ ∈ [−∆t ; ∆t].
This function is obviously even and satisfy:d2Qn
dτ2(τ) = −AQn(τ),
Qn(0) = Y (n∆t),dQn
dτ(0) = 0,
After having solved this equation, Y ((n + 1)∆t) can be computedusing Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t)
![Page 80: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/80.jpg)
Auxiliary Function
At each time step n we define an auxiliary function
Qn(τ) =Y (n∆t − τ) + Y (n∆t + τ)
2
for τ ∈ [−∆t ; ∆t].
This function is obviously even and satisfy:d2Qn
dτ2(τ) = −AQn(τ),
Qn(0) = Y (n∆t),dQn
dτ(0) = 0,
After having solved this equation, Y ((n + 1)∆t) can be computedusing Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t)
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Two different ways to solve the auxiliary equation
First Way
Solve d2Qn
dτ2(τ) = −AQn(τ),
Qn(0) = Y (n∆t),dQn
dτ(0) = 0,
by a fourth order modified equation scheme of time step ∆t/p andcompute Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t).
Remark
Is is equivalent to solve the original equation by a fourth ordermodified equation scheme of time step ∆t/p.
![Page 82: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/82.jpg)
Two different ways to solve the auxiliary equation
First Way
Solve d2Qn
dτ2(τ) = −AQn(τ),
Qn(0) = Y (n∆t),dQn
dτ(0) = 0,
by a fourth order modified equation scheme of time step ∆t/p andcompute Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t).
Remark
Is is equivalent to solve the original equation by a fourth ordermodified equation scheme of time step ∆t/p.
![Page 83: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/83.jpg)
Two different ways to solve the auxiliary equation
Second Way
Use a fourth order approximation of AQn(τ):
AQn(τ) ≈ AQn(0) +τ2
2Ad2Qn(0)
dτ2= AY (n∆t)− τ2
2AAY (n∆t),
then solved2Qn
dτ2(τ) = −AY (n∆t) +
τ2
2AAY (n∆t),
Qn(0) = Y (n∆t),dQn
dτ(0) = 0,
by a fourth order modified equation scheme of time step ∆t/pand compute Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t).
Remark
Is is equivalent to solve the original equation by a fourth ordermodified equation scheme of time step ∆t, whatever is p.
![Page 84: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/84.jpg)
Two different ways to solve the auxiliary equation
Second Way
Use a fourth order approximation of AQn(τ):
AQn(τ) ≈ AQn(0) +τ2
2Ad2Qn(0)
dτ2= AY (n∆t)− τ2
2AAY (n∆t),
then solved2Qn
dτ2(τ) = −AY (n∆t) +
τ2
2AAY (n∆t),
Qn(0) = Y (n∆t),dQn
dτ(0) = 0,
by a fourth order modified equation scheme of time step ∆t/pand compute Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t).
Remark
Is is equivalent to solve the original equation by a fourth ordermodified equation scheme of time step ∆t, whatever is p.
![Page 85: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/85.jpg)
Local Time-Stepping
d2Qn
dτ2+ AQn = 0
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Local Time-Stepping
d2Qn
dτ2+ AQn = 0
Let us now split Qn in two parts :
Qn =
[Qcoarse
n
Qfinen
]
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Local Time-Stepping
d2Qn
dτ2+ AQn = 0
Let us now split Qn in two parts :
Qn =
[Qcoarse
n
0
]+
[0
Qfinen
]
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Local Time-Stepping
d2Qn
dτ2+ AQn = 0
Let us now split Qn in two parts :
Qn =
[Qcoarse
n
0
]+
[0
Qfinen
]= (I − P)Qn + PQn, with P2 = P
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Local Time-Stepping
d2Qn
dτ2+ A(I − P)Qn + APQn = 0
Let us now split Qn in two parts :
Qn =
[Qcoarse
n
0
]+
[0
Qfinen
]= (I − P)Qn + PQn, with P2 = P
![Page 90: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/90.jpg)
Local Time-Stepping
d2Qn
dτ2+ A(I − P)Qn + APQn = 0
Idea
Approximate only A(I − P)Qn(τ) by
A(I − P)Qn(τ) ≈ A(I − P)Qn(0) +τ2
2A(I − P)
d2Qn(0)
dτ2
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Local Time-Stepping
d2Qn
dτ2+ A(I − P)Qn + APQn = 0
Idea
Approximate only A(I − P)Qn(τ) by
A(I − P)Qn(τ) ≈ A(I − P)Y (n∆t)− τ2
2A(I − P)AY (n∆t)
![Page 92: Efficient Space and Time Discretization of the Wave Equation ......E cient Space and Time Discretization of the Wave Equation. Application to the Reverse Time Migration J. Diaz1;2](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f912cd48ca08e0cb2091664/html5/thumbnails/92.jpg)
Local Time-Stepping
d2Qn
dτ2+ A(I − P)Qn + APQn = 0
Idea
Approximate only A(I − P)Qn(τ) by
A(I − P)Qn(τ) ≈ A(I − P)Y (n∆t)− τ2
2A(I − P)AY (n∆t)
So that Qn is the solution to∣∣∣∣∣∣∣∣∣d2Qn(τ)
dτ2+ A(I − P)Y (t)− τ2
2A(I − P)AY (n∆t) + APQn(τ) = 0
Qn(0) = Y (t)
Q ′n(0) = 0
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Algorithm of the fourth order local time-stepping scheme
Computation of Q(∆t)
We solve
∣∣∣∣∣∣∣∣∣d2
dτ2Qn(τ) + A(I − P)Y n − τ2
2A(I − P)AY n + APQn(τ) = 0
Qn(0) = Y n
Q ′n(0) = 0from τ = 0 to τ = ∆t, using a fourth order Modified Equation Scheme
with a time step∆t
p.
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Algorithm of the fourth order local time-stepping scheme
Computation of Q(∆t)
Q0n = Y n
V 1 = −A(I − P)Y n − APQ0n = −AY n
V 2 = A(I − P)AY n − APV 1
Q1pn = Q0
n +∆t2
2p2V 1 +
∆t4
24p4V 2
For i = 1..p − 1
V 1 = −A(I − P)Y n + 12
(i∆tp
)2A(I − P)AY n − APQ
ipn
V 2 = A(I − P)AY n − APV 1
Qi+1p
n = 2Qipn − Q
i−1p
n +∆t2
p2V 1 +
∆t4
12p4V 2
Endfor