报告人：Michael Dumbser（意大利特伦托大学、南方科技大学）

时 间：2026年10月29日16:30

地 点：海韵园行政楼C503

内容摘要:

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.  

个人简介：

Prof. Michael Dumbser is a Full Professor of Numerical Analysis in the Department of Civil, Environmental and Mechanical Engineering at the University of Trento, Italy, where he heads the Laboratory of Applied Mathematics. He obtained his doctorate in engineering from the University of Stuttgart in 2005 and subsequently joined the University of Trento, where he served as an Associate Professor before becoming a Full Professor. His research focuses on high-order numerical methods for hyperbolic partial differential equations, including ADER, discontinuous Galerkin, finite volume and structure-preserving schemes, with applications to computational fluid dynamics, solid mechanics, geophysical flows and relativistic astrophysics.  He serves as an Associate Editor of Journal of Computational Physics, Communications in Computational Physics and an Editor of the International Journal for Numerical Methods in Fluids, et al. He received the Heinz Maier-Leibnitz Prize and the Richard von Mises Prize in 2007, an ERC Starting Grant in 2011 and an ERC Advanced Grant in 2026. He has published more than 200 articles in top-ranked international journals, including SIAM Journal on Numerical Analysis,SIAM Journal on Scientific Computing, and Journal of Computational Physics, et al.

联系人：邱建贤