The effect of basis polynomial degree on the performance of compressible flow simulations employing a split-form DG method

Abstract

High order methods have garnered significant interest recently for simulating compressible flows due to their low dissipation nature. In this regard, the use of discontinuous Galerkin method has been actively pursued due to the advantages ensuing from its compact interpolation. However, these methods generally require employing a limiter for stable computation of supersonic compressible flows that contain shocks. Many limiters are designed to incorporate ideas from robust second order implementations of the Finite Volume Method for shock capturing. It is therefore prudent to examine the effectiveness of high order discontinuous Galerkin solutions when simulating flows containing discontinuities. To this end, the present work investigates the variation of accuracy per degrees of freedom as the basis polynomial degree is changed. Several standard one-dimensional and two-dimensional compressible flow cases are simulated for multiple basis polynomial degrees while employing different meshes to keep the degrees of freedom fixed. It is found that high order solutions are more efficient than second order solution as regards the accuracy per degrees of freedom. The improvement in this metric with increasing basis polynomial degree plateaus at cubic basis degree for cases containing strong discontinuities (shock Mach numbers greater than 3), and shifts to a further higher basis in simpler problems. However, in problems containing slowly moving shocks and or weak discontinuities, high order solutions generate increased numerical oscillations, suggesting the need for improvement in anti-aliasing techniques.