Vol. 6, No. 1, 2011

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

Peter McCorquodale and Phillip Colella

Vol. 6 (2011), No. 1, 1–25

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 methods, finite-volume methods, adaptive mesh refinement, hyperbolic partial differential equations
Mathematical Subject Classification 2000
Primary: 65M55
Received: 4 June 2010
Revised: 12 November 2010
Accepted: 28 January 2011
Published: 7 March 2011
Peter McCorquodale
Lawrence Berkeley National Laboratory
1 Cyclotron Road
MS 50A-1148
Berkeley CA 94720
United States
Phillip Colella
Applied Numerical Algorithms Group
Lawrence Berkeley National Laboratory
1 Cyclotron Road MS 50A-1148
Berkeley CA 94720
United States