An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping and p-adaptivity

SUMMARY This article describes the extension of the arbitrary high-order Discontinuous Galerkin (ADER-DG) method to treat locally varying polynomial degress 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 3-D 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 at the surface may be already obtained by the fine mesh. Therefore, it can be more adequate in some cases to use a lower order method in the small elements to reduce the CPU effort without loosing much accuracy. 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 of the computational domain. In contrast, by using local time stepping, each element can use its optimal time step given by the local stability condition. Our proposed LTS algorithm for ADER-DG is very general and does not need any temporal synchronization between the elements. Due to the ADER approach, accurate time interpolation is automatically provided at the element interfaces such that the computational overhead is very small and such that the method maintains the uniform high order of accuracy in space and time as in the usual ADER-DG schemes with a globally constant time step. However, the LTS ADER-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. This holds especially for unstructured tetrahedral meshes that contain strongly degenerate elements, so-called slivers. We show numerical convergence results and CPU times for LTS ADER-DG schemes up to sixth order in space and time on irregular tetrahedral meshes containing elements of very different size and also on tetrahedral meshes containing slivers. Further validation of the algorithm is provided by results obtained for the layer over half-space (LOH.1) benchmark problem proposed by the Pacific Earthquake Engineering Research Center. Finally, we present a realistic application on earthquake modelling and ground motion prediction for the alpine valley of Grenoble