High-order, finite-volume methods in mapped coordinates

Abstract

a b s t r a c tWe present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourthorder accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge-Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.

High-order central ENO finite-volume scheme for ideal MHD

“…Note that all these high-order procedures rely on mapping the governing fluid flow equations on spherical geometry to a Cartesian reference computational domain using non-orthogonal curvilinear coordinates for each sector of the cubed-sphere grid. In other work, Colella et al [30,31] have developed a high-order FVM on locally-structured grids and performed preliminary studies regarding the high-order interpolation of ghost cell values at the sector boundaries of cubed-sphere grids [30]. It appears that our work is the first to present a numerical scheme on 3D cubed-sphere grids with order of accuracy higher than two uniformly in all three dimensions.…”

A Freestream-Preserving High-Order Finite-Volume Method for Mapped Grids with Adaptive-Mesh Refinement

“…Previous related efforts involved finding a method for ensuring freestream preservation on mapped grids 4 and designing a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement. 1 Important details from these previous works are presented in this section.…”

“…It is important to design methods that guarantee freestream preservation, a property that ensures a uniform flow is not affected by the choice of mapping and discretization. A high-order method has been developed by Colella et al 4 that retains the freestream preservation property at any order of accuracy on mapped grids.…”

“…In this work, we merge the fourth-order AMR work of McCorquodale and Colella 1 for Cartesian grids with the freestream-preserving technology on mapped grids of Colella et al 4 To maintain conserved quantities as the grid levels appear, disappear, and migrate within the computational domain following solution features, we adopt the procedure described by Bell et al 5 In that procedure, special adjustments, also described herein, precede a modification of the grid hierarchy to ensure that the transfer of solution information between grid levels is conservative. The combination of mapped grids with AMR requires additional treatment to ensure that the freestream-preservation property is retained.…”

High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids

“…The starting point for the present work is the high-order finite-volume method in Colella et al [10]. The advantage of this approach is that it is strongly conservative in the sense of [41,40], high-order accurate, and freestream-preserving.…”