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

Abstract

A fourth-order accurate finite-volume method is presented for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space combined with detailed mechanisms for accommodating the adapting grids ensure that conservation is maintained and that the divergence of a constant vector field is always zero (freestream-preservation property). Advancement in time is achieved with a fourth-order Runge-Kutta method.

An adaptive multiblock high-order finite-volume method for solving the shallow-water equations on the sphere

“…With adaptive mesh refinement (AMR), we extend the approach of [19] on singleblock mapped grids to the mapped-multiblock grids of the cubed sphere. What makes the cubed sphere different from single-block mapped grids is that the solution is on a manifold, we are able to use analytic formulae for integrals of J , and adjacent panels have different mappings.…”

“…The approach in this paper is based on the finite-volume mapped-grid technology in [10], which is extended to work with AMR in [19]. We apply these methods on cubed-sphere meshes, which consist of six panels with a separate mapping on each panel.…”

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

“…These issues have been addressed for the single mapped-coordinate case in [14]. We also note that many of the methods for interpolation of ghost cells at refinement boundaries in adaptive mesh refinement do not use limiting for that process (e.g.…”

“…We also note that many of the methods for interpolation of ghost cells at refinement boundaries in adaptive mesh refinement do not use limiting for that process (e.g. [1,25,14]) while still leading to robust simulations of shocks. The limiters and other dissipation mechanisms used in computing the fluxes from the ghost cell data are sufficient to obtain robust calculations of discontinuities, and it is likely that the same will be the case for interpolating ghost cells at block boundaries.…”

APSARA: A multi-dimensional unsplit fourth-order explicit Eulerian hydrodynamics code for arbitrary curvilinear grids

“…They demonstrated the capability of their approach to solve an elliptic equation and a scalar, linear hyperbolic equation up to fourth-order accuracy. An extension of the method to solve nonlinear hyperbolic equations was presented for Cartesian coordinates by McCorquodale & Colella (2011) and for mapped coordinates by Guzik et al (2012). This extension is nontrivial because it is necessary to perform nonlinear transformations between point and (zone and face) averaged values of the conserved variables and fluxes ensuring fourthorder accuracy.…”