2019/10/14 by Kenneth Duru, Duru, Kenneth, Leonhard Rannabauer +7
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Physics (physics.comp-ph) #Distributed #Electromagnetic Simulation and Numerical Methods #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Geophysics (physics.geo-ph) #Numerical Analysis (math.NA) #Numerical methods for differential equations #Parallel #and Cluster Computing (cs.DC)
paper · pdf · doi:10.48550/arxiv.1910.06477
openalex publication_date 2019/10/14 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We present a stable discontinuous Galerkin (DG) method with a perfectly\nmatched layer (PML) for three and two space dimensional linear elastodynamics,\nin velocity-stress formulation, subject to well-posed linear boundary\nconditions. First, we consider the elastodynamics equation, in a cuboidal\ndomain, and derive an unsplit PML truncating the domain using complex\ncoordinate stretching. Leveraging the hyperbolic structure of the underlying\nsystem, we construct continuous energy estimates, in the time domain for the\nelastic wave equation, and in the Laplace space for a sequence of PML model\nproblems, with variations in one, two and three space dimensions, respectively.\nThey correspond to PMLs normal to boundary faces, along edges and in corners.\nSecond, we develop a DG numerical method for the linear elastodynamics equation\nusing physically motivated numerical flux and penalty parameters, which are\ncompatible with all well-posed, internal and external, boundary conditions.\nWhen the PML damping vanishes, by construction, our choice of penalty\nparameters yield an upwind scheme and a discrete energy estimate analogous to\nthe continuous energy estimate. Third, to ensure numerical stability of the\ndiscretization when PML damping is present, it is necessary to extend the\nnumerical DG fluxes, and the numerical inter-element and boundary procedures,\nto the PML auxiliary differential equations. This is crucial for deriving\ndiscrete energy estimates analogous to the continuous energy estimates. By\ncombining the DG spatial approximation with the high order ADER time stepping\nscheme and the accuracy of the PML we obtain an arbitrarily accurate wave\npropagation solver in the time domain. Numerical experiments are presented in\ntwo and three space dimensions corroborating the theoretical results.\n