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.