A high-order discontinuous Galerkin method for compressible flows with immersed boundaries

We present a higher order discretization scheme for the compressible Euler and Navier-Stokes equations with immersed boundaries. Our approach makes use of a discontinuous Galerkin discretization in a domain that is implicitly defined by means of a level set function. The zero iso-contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary-fitted cells. In order to retain the full order of convergence of the scheme, it is crucial to perform volume and surface integrals in boundary cells with high accuracy. This is achieved using a linear moment-fitting strategy. Moreover, we apply a non-intrusive cell-agglomeration technique that averts problems with very small and ill-shaped cuts. The robustness, accuracy, and convergence properties of the scheme are assessed in several two-dimensional test cases for the steady compressible Euler and Navier-Stokes equations. Approximation orders range from 0 to 4, even though the approach directly generalizes to even higher orders. In all test cases with a sufficiently smooth solution, the experimental order of convergence matches the expected rate for discontinuous Galerkin schemes. Copyright (c) 2016 John Wiley & Sons, Ltd.