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Motivation for mathematicians

Coupling of the Discontinuous Galerkin and Finite Difference techniques to simulate seismic waves in the presence of sharp interfaces J. Diaz , D. Kolukhin , V. Lisitsa , V. Tcheverda. Motivation for mathematicians. Free-surface perturbation. σ = 1.38, I = 44.9 м.

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Motivation for mathematicians

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  1. Coupling of the Discontinuous Galerkin and Finite Difference techniques to simulate seismic waves in the presence of sharp interfacesJ. Diaz, D. Kolukhin, V. Lisitsa, V. Tcheverda

  2. Motivation for mathematicians Free-surface perturbation σ = 1.38, I = 44.9 м

  3. Motivation for mathematicians Free-surface perturbation σ = 1.38, I = 44.9 м 30%

  4. Motivation for geophysicists

  5. Motivation for geophysicists

  6. Motivation for geophysicists Original source

  7. Motivation for geophysicists Diffraction of Rayleigh wave, secondary sources

  8. Motivation

  9. Standard staggered grid scheme

  10. Standard staggered grid scheme • Easy to implement • Able to handle complex models • High computational efficiency • Suitable accuracy • Poor approximation of sharp interfaces

  11. Discontinuous Galerkin method Elastic wave equation in Cartesian coordinates:

  12. Discontinuous Galerkin method

  13. Discontinuous Galerkin method

  14. Discontinuous Galerkin method • Use of polyhedral meshes • Accurate description of sharp interfaces • Hard to implement for complex models • Computationally intense • Strong stability restrictions (low Courant numbers)

  15. Dispersion analysis (P1) Courant ratio 0.25

  16. Dispersion analysis (P2) Courant ratio 0.144

  17. Dispersion analysis (P3) Courant ratio 0.09

  18. DG + FD • Finite differences: • Easy to implement • Able to handle complex models • High computational efficiency • Suitable accuracy • Poor approximation of sharp interfaces • Discontinuous Galerkin method: • Use of polyhedral meshes • Accurate description of sharp interfaces • Hard to implement for complex models • Computationally intense • Strong stability restrictions (low Courant numbers)

  19. A sketch P1-P3 DG on irregular triangular grid to match free-surface topography P0 DG on regular rectangular grid = conventional (non-staggered grid scheme) – transition zone Standard staggered grid scheme

  20. Experiments DG

  21. FD+DG on rectangular grid P0 DG on regular rectangular grid Standard staggered grid scheme

  22. Spurious Modes 2D example in Cartesian coordinates

  23. Spurious Modes 2D example in Cartesian coordinates

  24. Interface Reflected waves Incident waves Transmitted artificial waves Transmitted true waves

  25. Conjugation conditions Reflected waves Incident waves Transmitted artificial waves Transmitted true waves

  26. Experiments

  27. Numerical experiments Source P S Surface Xs=4000, Zs=110 (10 meters below free surface), volumetric source, freq=30Hz Zr=5 meters below free surface Vertical component is presented

  28. Comparison with FD DG P1 h=2.5 m. FD h=2.5 m. The same amplitude normalization Numerical diffraction

  29. Comparison with FD DG P1 h=2.5 m. FD h=1m. The same amplitude normalization Numerical diffraction

  30. Numerical experiments Xs=4500, Zs= 5 meters below free surface, volumetric source, freq=20Hz Zr=5 meters below free surface

  31. Numerical Experiments

  32. Numerical Experiment – Sea Bed Source position x=12,500 m, z=5 m Ricker pulse with central frequency of 10 Hz Receivers were placed at the seabed.

  33. Numerical Experiments

  34. Conclusions • Discontinuous Galerkin method allows properly handling wave interaction with sharp interfaces, but it is computationally intense • Finite differences are computationally efficient but cause high diffractions because of stair-step approximation of the interfaces. • The algorithm based on the use of the DG in the upper part of the model and FD in the deeper part allows properly treating the free surface topography but preserves the efficiency of FD simulation.

  35. Thank you for attention

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