A stable discontinuous Galerkin method for linear elastodynamics in geometrically complex media using physics based numerical fluxes
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High order accurate and explicit time-stable solvers are well suited for hyperbolic wave propagation problems. For the complexities of real geometries, internal interfaces, nonlinear boundary and interface conditions, discontinuities and sharp wave fronts become fundamental features of the solutions. These are also effects of the presence of disparate spatial and temporal scales, present in real media and sources. As a result high order accuracy, geometrically flexible and adaptive numerical algorithms are critical for high fidelity and efficient simulations of wave phenomena in many applications. Using a physics-based numerical penalty-flux, we develop a provably energy-stable discontinuous Galerkin approximation of the elastic wave equation in complex and discontinuous media. By construction, our numerical flux is upwind and yields a discrete energy estimate analogous to the continuous energy estimate. The discrete energy estimate holds for conforming and non-conforming curvilinear elements. The ability to handle non-conforming curvilinear meshes allows for flexible adaptive mesh refinement strategies. The numerical scheme has been implemented in ExaHyPE, a simulation engine for hyperbolic PDEs on adaptive Cartesian meshes, for exascale supercomputers. We present 3D numerical experiments demonstrating stability and high order accuracy. Finally, we present a regional geophysical wave propagation problem in a 3D Earth model with geometrically complex free-surface topography.
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