Speaker：Michael Dumbser (University of Trento & SUSTech)

Time：2026-10-29 16: 30

Location：Conference Room C503 at Administration Building at Haiyun Campus

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, which are obtained by cutting embedded boundaries out of a regular Cartesian background mesh. As a result, Cartesian cut cell meshes may contain cut cells of arbitrarily small size, which leads to the well-known small cell problem, i.e. the classical CFL-type stability condition of explicit schemes would impose a severe restriction on the admissible time step length on small cut cells. 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, while the zone close to the embedded boundary that consists of cut cells and their direct face neighbors is discretized with an implicit space-time DG scheme. Both schemes are derived from the same weak formulation of the PDE in space-time and employ discontinuous space-time polynomials of equal degree $N$, namely the local space-time predictor of the ADER-DG scheme on the explicit cells and the space-time DG solution of the implicit cells. Both schemes are coupled with each other directly via the numerical flux. The resulting mixed explicit-implicit scheme is conservative by construction, as both schemes share the same numerical flux, and its time step is determined only by the size of the Cartesian background cells, since the small cut cells and their direct neighbors are stabilized by the proposed zonally implicit space-time DG discretization. The element-local space-time DG formulation of the explicit fully-discrete ADER-DG schemes makes the coupling to the zonally implicit space-time DG method very natural. Curved embedded boundaries are represented by high order isoparametric sub-triangles inside each cut cell, and the nonlinear system of the implicit cells around the embedded boundaries is efficiently solved via an inexact matrix-free Newton--Krylov method with a simple block Jacobi preconditioner. For problems with shock waves, we add a suitable artificial viscosity (AV), which is discretized with a non-symmetric interior penalty Galerkin (NIPG) method controlled by an oscillation-eliminating smoothness sensor. The numerical results show that the scheme essentially reaches its nominal order of accuracy N+1 for smooth solutions and for polynomial approximation degrees up to N=5, on meshes with minimal volume fractions of the small cut cells down to the order of 1e-7.