Spectral difference method with a posteriori limiting: application to the Euler equations in one and two space dimensions

We present a new numerical scheme which combines the spectral difference (SD) method up to arbitrary high order with a-posteriori limiting using the classical MUSCL-Hancock scheme as fallback scheme. It delivers very accurate solutions in smooth regions of the flow while capturing sharp discontinuities without spurious oscillations. We exploit the strict equivalence between the SD scheme and a finite-volume scheme based on the SD control volumes to enable a straightforward limiting strategy. At the end of each stage of our high-order time-integration ADER (Arbitrary high order using Derivatives) scheme, we check if the high-order solution is admissible under a number of numerical and physical criteria. If not, we replace the high-order fluxes of the troubled cells by fluxes from our robust second-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) fallback scheme. We apply our method to a suite of test problems for the one-dimensional and two-dimensional Euler equations. We demonstrate that this combination of SD and ADER provides a virtually arbitrary high order of accuracy while at the same time preserving good sub-element shock capturing capabilities.