A high-order finite-volume method for conservation laws on locally refined grids

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

“…We therefore use least-squares interpolation to transfer cell-average information between the overlapping, curvilinear grids. Least-squares interpolation has also been used at block boundaries for embedded-boundary methods in [20,21] and for high-order coarse-fine mesh interpolation in AMR [25]. The methods of the present paper are applied in [26] to the solution of the shallow-water equations on the surface of a sphere, using AMR.…”

“…These exchange operations are based on solving an overdetermined system of equations by the method of least squares, as in the spatial coarse-fine interpolation in [25]. For the purposes of presentation, this description is specialized to fourth-order interpolation but can be easily generalized to arbitrarily higher order.…”

“…with B ρ as defined in (17). Since v is uniform, as in the example of Section 4.2, the exact solution of (15) is again given by (25). Results of a grid convergence study are presented in Table 3, which shows the L ∞ norm of the difference between calculated and exact solutions for different resolutions.…”

“…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.…”

A freestream-preserving fourth-order finite-volume method in mapped coordinates with adaptive-mesh refinement

“…Colella et al [1] have developed a high-order method that retains the freestream preservation property at any order of accuracy on mapped grids. In this work, we merge the fourthorder AMR work of McCorquodale and Colella [2] for Cartesian grids with the freestream-preserving technology on mapped grids of Colella et al [1] to achieve a fourth-order, freestream-preserving, finite-volume method on mapped grids with AMR. 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 [3].…”