Motivation for mathematicians

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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. - PowerPoint PPT Presentation

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  • 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* = 1.38, I = 44.9 Free-surface perturbation

  • Motivation for mathematicians* = 1.38, I = 44.9 Free-surface perturbation30%

  • Motivation for geophysicists*

  • *Motivation for geophysicists

  • Motivation for geophysicists*Original source

  • Motivation for geophysicists*Diffraction of Rayleigh wave, secondary sources

  • Motivation*

  • Standard staggered grid scheme*

  • Standard staggered grid scheme*Easy to implementAble to handle complex modelsHigh computational efficiencySuitable accuracyPoor approximation of sharp interfaces

  • Discontinuous Galerkin methodElastic wave equation in Cartesian coordinates:

  • Discontinuous Galerkin method

  • Discontinuous Galerkin method

  • Discontinuous Galerkin methodUse of polyhedral meshesAccurate description of sharp interfacesHard to implement for complex modelsComputationally intenseStrong 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*Finite differences:Easy to implementAble to handle complex modelsHigh computational efficiencySuitable accuracyPoor approximation of sharp interfaces

    Discontinuous Galerkin method:Use of polyhedral meshesAccurate description of sharp interfacesHard to implement for complex modelsComputationally intenseStrong stability restrictions (low Courant numbers)

  • A sketchP1-P3 DG on irregular triangular grid to match free-surface topographyP0 DG on regular rectangular grid = conventional (non-staggered grid scheme) transition zoneStandard staggered grid scheme*

  • Experiments*DG

    PPWReflection15~3 %30~0.5 %60~0.1 %120??? %

  • FD+DG on rectangular gridP0 DG on regular rectangular gridStandard staggered grid scheme*

  • Spurious Modes2D example in Cartesian coordinates

  • Spurious Modes2D example in Cartesian coordinates

  • InterfaceIncident wavesReflected wavesTransmitted artificial wavesTransmitted true waves

  • Conjugation conditionsIncident wavesReflected wavesTransmitted artificial wavesTransmitted true waves

  • Experiments*

    PPWReflection151.6 %300.5 %600.1 %1200.03 %

  • Numerical experiments*PSSurfaceXs=4000, Zs=110 (10 meters below free surface), volumetric source, freq=30HzZr=5 meters below free surfaceVertical component is presentedSource

  • Comparison with FD*DG P1 h=2.5 m.FD h=2.5 m.The same amplitude normalizationNumerical diffraction

  • Comparison with FD*DG P1 h=2.5 m.FD h=1m.The same amplitude normalizationNumerical diffraction

  • Numerical experiments*Xs=4500, Zs= 5 meters below free surface, volumetric source, freq=20HzZr=5 meters below free surface

  • Numerical Experiments*

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

  • Numerical Experiments

  • ConclusionsDiscontinuous Galerkin method allows properly handling wave interaction with sharp interfaces, but it is computationally intenseFinite 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*