Immersed boundaries in the discontinuous Galerkin spectral element method through hp-adaptivity

Abstract

The immersed boundary method is a promising numerical technique that allows for modeling of complex geometries without the need for body conforming meshes. However, immersed boundary methods present a significant reduction in accuracy. In this paper, we implement the volume penalty method in an hp-adaptive discontinuous Galerkin spectral method framework to solve the two-dimensional acoustic wave equation with immersed boundaries. We demonstrate that combining low porosity, which represents the immersed boundary, with hp-adaptivity reduces oscillations, localizes error to the vicinity of the immersed boundary and improves the overall accuracy. A variety of test cases are presented to show that the implementation is capable of modeling wave propagation in complex geometries with simple Cartesian grids.