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

*(English)*Zbl 1252.65163Summary: 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 P. Colella et al. [J. Comput. Phys. 230, No. 8, 2952–2976 (2011; Zbl 1218.65119)] 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 P. Colella and M. D. Sekora [J. Comput. Phys. 227, No. 15, 7069–7076 (2008; Zbl 1152.65090)], 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.

##### MSC:

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |