Stabilized discontinuous Galerkin methods for solving hyperbolic conservation laws on grids with embedded objects

Abstract

This thesis covers a novel penalty stabilization for solving hyperbolic conservation laws using discontinuous Galerkin methods on grids with embedded objects. We consider cut cell grids, that are constructed by cutting the given object out of a Cartesian background grid. The resulting cut cells require special treatments, e.g., adding stabilization terms. In the context of hyperbolic conservation laws, one has to overcome the small cell problem: standard explicit time stepping becomes unstable on small cut cells when the time step is selected based on larger background cells. This work will present the Domain of Dependence (DoD) stabilization in one and two dimensions. By transferring additional information between the small cut cell and its neighbors, the DoD stabilization restores the correct domains of dependence in the neighborhood of the cut cell. The stabilization is added as penalty terms to the semi-discrete scheme. When combined with a standard explicit time-stepping scheme, the stabilized scheme remains stable for a time-step length based on the Cartesian background cells. Thus, the small cell problem is solved. In the first part of this work, we will consider one-dimensional hyperbolic conservation laws. We will start by explaining the ideas of the stabilization for linear scalar problems before moving to non-linear problems and systems of hyperbolic conservation laws. For scalar problems, we will show that the scheme ensures monotonicity when using its first-order version. Further, we will present an L2 stability result. We will conclude this part with numerical results that confirm stability and good accuracy. These numerical results indicate that for both, linear and non-linear problems, the convergence order in various norms for smooth tests is p+1 when using polynomials of degree p. In the second part, we will present first ideas for extending the DoD stabilization to two dimensions. We will consider different simplified model problems that occur when using two-dimensional cut cell meshes. An essential step for the extension to two dimensions will be the construction of weighting factors that indicate how we couple the multiple cut cell neighbors with each other. The monotonicity and L2 stability of the stabilized system will be confirmed by transferring the ideas of the proof from one to two dimensions. We will conclude by presenting numerical results for advection along a ramp, demonstrating convergence orders of p+1/2 to p+1 for polynomials of degree p. Additionally, we present preliminary results for the two-dimensional Burgers and Euler equations on model meshes.