High-Order Cut-Cell Discontinuous Galerkin Difference Discretization

Abstract

We present a high-order cut-cell method based on the discontinuous Galerkin difference (DGD) discretization. We leverage the inherent properties of the DGD basis functions to construct a cut-cell discretization that does not require special treatment to mitigate the small-cell problem. The paper describes how the DGD discretization can be constructed from an existing discontinuous Galerkin (DG) discretization, and we highlight differences between the DG and DGD methods. By performing condition-number studies on one- and two-dimensional model problems, we demonstrate that cut-cell DGD discretization remains well conditioned even when the cut-cell volume is orders of magnitude smaller than neighboring cells. We verify the high-order accuracy of the discretization by solving the two-dimensional steady-state Euler equations.