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

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. • hp-adaptivity reduces oscillations and dispersion error in immersed boundary method. • hp-adaptivity allows a smaller porosity in immersed boundary volume penalization. • hp-adaptivity uses a fraction of the computing time for high order immersed boundaries. • The immersed boundary method used with hp-adaptivity greatly simplifies meshing. • hp-adaptive Discontinuous Galerkin Spectral Elements can handle immersed boundaries.