HIGH-ORDER ACCURATE EMBEDDED-BOUNDARY DISCONTINUOUS GALERKIN METHODS FOR INVISCID GAS DYNAMICS

Abstract

This work presents a computational framework for solving the equations of inviscid gas dynamics over embedded geometries based on the discontinuous Galerkin (DG) method. The novelty of the framework is the ability to achieve high-order accuracy in the regions of smooth flow and to handle the presence of solution discontinuities via suitably introduced damping terms, which allow controlling spurious oscillations that are typical of high-order methods for first-order hyperbolic PDEs. The framework employs block structured Cartesian grids where a level set function defines implicitly the considered geometry. The domain is partitioned by intersecting the grid and the level set function, such that the resulting mesh consists of a collection of standard d-dimensional rectangular cells and a relatively smaller number of irregular cut cells in proximity of the boundary of the embedded geometry, with a simple cell merging strategy handling the presence of overly small cut cells. The DG formulation is used to discretize the governing equations in space and to introduce the damping terms. Runge-Kutta algorithms are then employed to integrate the resulting semi-discrete equations in time. Numerical tests are presented to show the high-order accuracy and the shock-capturing capabilities of the proposed approach.