A p-Adaptive Discontinuous Galerkin Method with Local Time Steps for Computational Seismology

Abstract

This article describes the application of a recently developed arbitrary high order accurate Discontinuous Galerkin (DG) method on High-Performance Computing facilities to treat locally varying polynomial degrees of the basis functions, so-called p-adaptivity, as well as locally varying time steps that may be different from one element to another. The p-adaptive version of the scheme is useful in complex three-dimensional models with small-scale features which have to be meshed with reasonably small elements to capture the necessary geometrical details of interest. Using a constant high polynomial degree of the basis functions in the whole computational domain can lead to an unreasonably high CPU effort, since good spatial resolution may be already obtained by the fine mesh. To further increase computational efficiency, we present a new local time stepping (LTS) algorithm. For usual explicit time stepping schemes the element with the smallest time step resulting from the stability criterion of the method will dictate its time step to all the other elements. In contrast, local time stepping allows each element to use its optimal time step given by the local stability condition. A time interpolation is automatically provided at the element interfaces such that the computational overhead is negligible and such that the method maintains the uniform high order of accuracy in space and time as in the usual DG schemes with a globally constant time step. However, the LTS DG method is computationally much more efficient for problems with strongly varying element size or material parameters since it allows to reduce the total number of element updates considerably. We present a realistic application on earthquake modeling and ground motion prediction for the alpine valley of Grenoble.