ADER-ZIST - a mixed explicit ADER-DG and zonally implicit space-time DG approach for Cartesian cut cell meshes

Abstract: We present high order accurate mixed explicit-implicit discontinuous Galerkin (DG) schemes for the solution of hyperbolic systems of conservation laws on Cartesian cut cell meshes. Such meshes may contain arbitrarily small cut cells, which leads to the well-known small cell problem: the CFL condition of explicit schemes would severely restrict the admissible time step. To overcome this difficulty, we use high order explicit fully-discrete one-step ADER-DG schemes only on Cartesian cells away from the embedded boundaries. The zone close to the embedded boundary, which consists of cut cells and their direct face neighbors, is discretized with an implicit space-time DG scheme that is coupled to the explicit scheme directly via the numerical flux. The resulting scheme, called ADER-ZIST (ADER-DG with Zonally Implicit Space-Time DG), is conservative by construction, and its time step is determined only by the size of the Cartesian background cells. Curved embedded boundaries are represented by high order isoparametric sub-triangles inside the cut cells, and the nonlinear system of the implicit cells is solved via an inexact matrix-free Newton-Krylov method. For problems with shock waves, we add an artificial viscosity term whose strength is controlled by an oscillation-eliminating smoothness sensor. ADER-ZIST is assessed on a comprehensive set of test problems for the compressible Euler equations, ranging from smooth flows around straight and curved embedded boundaries to acoustic scattering and shock-dominated flows. The scheme essentially reaches its nominal order of accuracy N+1 for smooth solutions and for polynomial degrees up to N=5, on meshes with minimal volume fractions of the small cut cells down to the order of 10^-7.