Seminar in Numerical Analysis: Kenneth Duru (Ludwig-Maximilians-Universität München)
High order accurate and explicit time-stable solvers are well suited for hyperbolic wave propagation problems. However, because of the complexities of real geometries, internal interfaces, nonlinear boundary/interface conditions and the presence of disparate spatial and temporal scales present in real media and sources, discontinuities and sharp wave fronts become fundamental features of the solutions. Thus, high order accuracy, geometrically flexible and adaptive numerical algorithms are critical for high fidelity and efficient simulations of wave phenomena in many applications. I will present a physics-based numerical flux suitable for inter-element and boundary conditions in discontinuous Galerkin approximations of first order hyperbolic PDEs. Using this physics-based numerical penalty-flux, we will develop a provably energy-stable discontinuous Galerkin approximations of the elastic waves in complex and discontinuous media. By construction the numerical flux is upwind and yields a discrete energy estimate analogous to the continuous energy estimate. The discrete energy estimates hold 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 have been implemented in ExaHyPE, a simulation engine for hyperbolic PDEs on adaptive structured meshes, for exa-scale supercomputers. I will show 3D numerical experiments demonstrating stability and high order accuracy. Finally, we present a large scale geophysical regional wave propagation problem in a heterogeneous Earth model with geologically constrained media heterogeneity and geometrically complex free-surface topography.
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