High-order finite-volume adaptive methods on locally rectangular grids

We are developing a new class of finite-volume methods on locally-refined and mapped grids, which are at least fourth-order accurate in regions where the solution is smooth. This paper discusses the implementation of such methods for time-dependent problems on both Cartesian and mapped grids with adaptive mesh refinement. We show 2D results with the Berger–Colella shock-ramp problem in Cartesian coordinates, and fourth-order accuracy of the solution of a Gaussian pulse problem in a polytropic gas in mapped coordinates. 1. High-order finite-volume methods on Cartesian grids In the finite-volume approach, the spatial domain in R is discretized as a union of control volumes that covers the domain. With Cartesian grids, a control volume Vi takes the form Vi = [ih, (i+ u)h], i ∈ Z ,u = (1, 1, . . . , 1), where h is the grid spacing. A finite-volume discretization of a partial differential equation is based on averaging that equation over control volumes, applying the divergence theorem to replace volume integrals by integrals over the boundary of the control volume, and approximating the boundary integrals by quadratures. For example, for time-dependent problems of the form ∂U ∂t +∇ · ~ F = 0 (1) the discretized solution in space is the average of U over a control volume: 〈U〉i(t) = 1 hD ∫