Entropy Stable Weight-Adjusted Flux Reconstruction High-Order Method in Split Form for Compressible Flows on Curvilinear Grids

Abstract

The flux reconstruction method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the Discontinuous Galerkin (DG) method, on unstructured grids over complex geometries. Under a class of energy stable flux reconstruction (ESFR) schemes, the flux reconstruction method allows for larger time-steps than DG while ensuring stability for linear advection on linear elements. For nonlinear problems, split forms emerged as the popular approach proving stability for unsteady problems on coarse unstructured grids; with recent developments proving stability for the one-dimensional Burgers' equation through the incorporation of the ESFR correction functions on the volume terms. In curvilinear coordinates, the metric terms also add a nonlinearity to the scheme in reference space. This paper is the first to combine both the nonlinearity from the governing equation along with the nonlinearity from the curvilinear metric terms for nonlinearly stable flux reconstruction. Unfortunately, in curvilinear coordinates, the scheme requires that the dense mass matrix is inverted in every element. This paper incorporates a low-storage, weight-adjusted approach to approximate the inverse of the mass matrix, while preserving the desired nonlinear stability property. The theoretical results are verified with the inviscid Taylor-Green vortex problem on a coarse, non-symmetrically warped curvilinear grid.