A new discontinuous Galerkin spectral element method for elastic waves with physically motivated numerical fluxes

The discontinuous Galerkin spectral element method (DGSEM) is now an established method for computing approximate solutions of partial differential equations in many applications. Unlike continuous finite elements, in DGSEM, numerical fluxes are used to enforce inter-element conditions, and internal and external physical boundary conditions. This has been successful for many problems. However, for certain problems such as elastic wave propagation in complex media, and where several wave types and wave speeds are simultaneously present, a standard numerical flux may not be compatible with the physical boundary conditions. If surface or interface waves are present, this incompatibility may lead to numerical instabilities. We present a stable and arbitrary order accurate DGSEM for elastic waves with a physically motivated numerical flux. Our numerical flux is compatible with all well-posed, internal and external, boundary conditions, and can be easily extended to linear and nonlinear friction laws for modeling fracture in elastic solids and dynamic earthquake rupture processes. By construction our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. The spectral radius of the resulting spatial operator has an upper bound which is independent of the boundary and interface conditions, thus it is suitable for efficient explicit time integration. We present numerical experiments verifying high order accuracy and asymptotic numerical stability.

@article{id2445,
  author = {Duru, K. C.  and Rannabauer, Leonhard and Gabriel, Alice-Agnes and Igel, Heiner},
  doi = {10.1007/s10915-021-01565-1},
  journal = {Journal of Scientific Computing},
  language = {en},
  note = {open access version available at ArXiv https://arxiv.org/abs/1802.06380},
  number = {51},
  title = {A new discontinuous Galerkin spectral element method for elastic waves with physically motivated numerical fluxes},
  url = {https://doi.org/10.1007/s10915-021-01565-1},
  volume = {88},
  year = {2021},
}

%O Journal Article
%A Duru, K. C. 
%A Rannabauer, Leonhard
%A Gabriel, Alice-Agnes
%A Igel, Heiner
%R 10.1007/s10915-021-01565-1
%J Journal of Scientific Computing
%G en
%O open access version available at ArXiv https://arxiv.org/abs/1802.06380
%N 51
%T A new discontinuous Galerkin spectral element method for elastic waves with physically motivated numerical fluxes
%U https://doi.org/10.1007/s10915-021-01565-1
%V 88
%D 2021