Michael Dumbser

TL;DR: A conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method, which yields space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms that can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes.

TL;DR: A new WENO reconstruction technique is proposed that does not reconstruct point-values but entire polynomials which can easily be evaluated and differentiated at any point and thus can be implemented very efficiently even for unstructured grids in three space dimensions.

TL;DR: The development of the highly accurate ADER–DG approach for tetrahedral meshes provides a numerical technique to approach 3-D wave propagation problems in complex geometry with unforeseen accuracy.

TL;DR: A discontinuous Galerkin (DG) method combined with the ideas of the ADER time integration approach to solve the elastic wave equation in heterogeneous media in the presence of externally given source terms with arbitrary high-order accuracy in space and time on unstructured triangular meshes is presented.

TL;DR: A quadrature-free essentially non-oscillatory finite volume scheme of arbitrary high order of accuracy both in space and time for solving nonlinear hyperbolic systems on unstructured meshes in two and three space dimensions is presented.