An efficient ADERDG local time stepping scheme for 3D HPC simulation of seismic waves in poroelastic media
Abstract
Many applications from the fields of seismology and geoengineering require simulations of seismic waves in porous media. Biot's theory of poroelasticity describes the coupling between solid and fluid phases and introduces a stiff reactive source term (Darcy's Law) into the elastodynamic wave equations, thereby increasing computational cost of respective numerical solvers and motivating efficient methods utilising HighPerformance Computing.
We present a novel realisation of the discontinuous Galerkin scheme with Arbitrary HighOrder DERivative time stepping (ADERDG) that copes with stiff source terms. To integrate this source term with a reasonable time step size, we utilise an elementlocal spacetime predictor, which needs to solve mediumsized linear systems  each with 1,000 to 10,000 unknowns  in each element update (i.e., billions of times). We present a novel blockwise backsubstitution algorithm for solving these systems efficiently, thus enabling largescale 3D simulations. In comparison to LU decomposition, we reduce the number of floatingpoint operations by a factor of up to 25, when using polynomials of degree 6. The blockwise backsubstitution is mapped to a sequence of small matrixmatrix multiplications, for which code generators are available to generate highly optimised code.
We verify the new solver thoroughly against analytical and semianalytical reference solutions in problems of increasing complexity. We demonstrate highorder convergence of the scheme for 3D problems. We verify the correct treatment of point sources and boundary conditions, including homogeneous and heterogeneous full space problems as well as problems with tractionfree boundary conditions. In addition, we compare against a finite difference solution for a newly defined 3D layer over halfspace problem containing an internal material interface and free surface. We find that extremely high accuracy is required to accurately resolve the slow, diffusive Pwave at a or near a free surface, while we also demonstrate that solid particle velocities are not affected by coarser resolutions. We demonstrate that by using a clustered local time stepping scheme, time to solution is reduced by a factor of 6 to 10 compared to global time stepping. We conclude our study with a scaling and performance analysis on the SuperMUCNG supercomputer, demonstrating our implementation's high computational efficiency and its potential for extremescale simulations.
 Publication:

Journal of Computational Physics
 Pub Date:
 April 2022
 DOI:
 10.1016/j.jcp.2021.110886
 arXiv:
 arXiv:2108.10565
 Bibcode:
 2022JCoPh.45510886W
 Keywords:

 Poroelasticity;
 Discontinuous Galerkin;
 Wave propagation;
 High Performance Computing (HPC);
 Computational seismology;
 ADERDG;
 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Mathematical Software;
 Physics  Computational Physics;
 Physics  Geophysics
 EPrint:
 37 pages, 18 figures, published in the Journal of Computational Physics