ADER schemes are recent finite volume methods for hyperbolic conservation laws, which can be viewed as generalizations of the classical first order Godunov method to arbitrary high orders. In the ADER approach, high order polynomial reconstruction from cell averages is combined with high order flux evaluation, where the latter is done by solving generalized Riemann problems across cell interfac...

[1] We introduce the application of an arbitrary high-order derivative (ADER) discontinuous Galerkin (DG) method to simulate earthquake rupture dynamics. The ADER-DG method uses triangles as computational cells which simplifies the process of discretization of very complex surfaces and volumes by using external automated tools. Discontinuous Galerkin methods are well suited for solving dynamic ...

Seismic simulations in realistic 3D Earth models require petaor even exascale computing power to capture small-scale features of high relevance for scientific and industrial applications. In this paper, we present optimizations of SeisSol – a seismic wave propagation solver based on the Arbitrary high-order accurate DERivative Discontinuous Galerkin (ADER-DG) method on fully adaptive, unstructu...

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge problem. Based on the three dimensional consistent unstructured tetrahedron meshes, we study the discontinuous Galerkin (DG) finite element method [1] combined with the adaptive mesh refinement (AMR) [2, 3] for solving Euler equations in this paper. That is according t...

We present the first high order one-step ADER-WENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resul...

In this paper, we present a time integration scheme applied to the Discontinuous Galerkin finite element method (DG-FEM, [1]) for the computation of electromagnetic fields in the interior of three-dimensional structures. This approach is also known as Arbitrary High-Order Derivative Discontinuous Galerkin (ADER-DG, [2, 3]). By this method, we reach arbitrary high accuracy not only in space but ...

In this article we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First, in order to achieve high order accuracy in space, a nonlinear weighted essentially non-oscillatory reconstruction procedure is applied to the cell averages at t...

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials degree N inside each element, in our approach discrete is represented by piecewise continuous within finite element basis defined subgrid polygon. call...