J. Diaz, D. Kolukhin, V. Lisitsa, V. Tcheverda 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 1

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

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

Motivation for geophysicists 4

5

6 Original source

Motivation for geophysicists 7 Diffraction of Rayleigh wave, secondary sources

Motivation 8

Standard staggered grid scheme 9

10 Easy to implement Able to handle complex models High computational efficiency Suitable accuracy Poor approximation of sharp interfaces

Discontinuous Galerkin method Elastic wave equation in Cartesian coordinates:

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)

Dispersion analysis (P1) Courant ratio 0.25

Dispersion analysis (P2) Courant ratio 0.144

Dispersion analysis (P3) Courant ratio 0.09

DG + FD 18 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)

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 19

Experiments 20 PPWReflection 15~3 % 30~0.5 % 60~0.1 % 120??? % DG

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

Spurious Modes 2D example in Cartesian coordinates

Spurious Modes 2D example in Cartesian coordinates

Interface Incident waves Reflected waves Transmitted artificial waves Transmitted true waves

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

Experiments 26 PPWReflection % % % %

Numerical experiments 27 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 Source

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

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

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

Numerical Experiments 31

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.

Numerical Experiments

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.

Thank you for attention 35