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

We present a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement.The underlying method is a generalization of that developed by Colella, Dorr, Hittinger and Martin (2009) to nonlinear systems, and is based on using fourth-order accurate quadratures for computing fluxes on faces, combined with fourth-order accurate Runge-Kutta discretization in time.To interpolate boundary conditions at refinement boundaries, we interpolate in time in a manner consistent with the individual stages of the Runge-Kutta method, and interpolate in space by solving a least-squares problem over a neighborhood of each target cell for the coefficients of a cubic polynomial.The method also uses a variation on the extremum-preserving limiter of Colella and Sekora (2008), as well as slope flattening and a fourth-order accurate artificial viscosity for strong shocks.We show that the resulting method is fourth-order accurate for smooth solutions, and is robust in the presence of complex combinations of shocks and smooth flows. High-order finite-volume methodsIn the finite-volume approach, the spatial domain in ‫ޒ‬ D is discretized as a union of rectangular control volumes that covers the spatial domain.For Cartesian-grid finite-volume methods, a control volume V i takes the formwhere 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, This work was supported by the Director, Office of Science, Office of Advanced Scientific Computing Research, of the U.S.