Communications in Applied Mathematics and Computational Science

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

Peter McCorquodale and Phillip Colella

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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.

Article information

Commun. Appl. Math. Comput. Sci., Volume 6, Number 1 (2011), 1-25.

Received: 4 June 2010
Revised: 12 November 2010
Accepted: 28 January 2011
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 65M55: Multigrid methods; domain decomposition

high-order methods finite-volume methods adaptive mesh refinement hyperbolic partial differential equations


McCorquodale, Peter; Colella, Phillip. A high-order finite-volume method for conservation laws on locally refined grids. Commun. Appl. Math. Comput. Sci. 6 (2011), no. 1, 1--25. doi:10.2140/camcos.2011.6.1.

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  • M. Barad and P. Colella, A fourth-order accurate local refinement method for Poisson's equation, J. Comput. Phys. 209 (2005), no. 1, 1–18.
  • J. B. Bell, P. Colella, J. A. Trangenstein, and M. Welcome, Adaptive mesh refinement on moving quadrilateral grids, Proceedings of the 9th AIAA Computational Fluid Dynamics Conference, AIAA, June 1989, pp. 471–479.
  • M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys. 82 (1989), no. 1, 64–84.
  • P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), no. 1, 171–200.
  • P. Colella, M. Dorr, J. Hittinger, and D. F. Martin, High-order, finite-volume methods in mapped coordinates, J. Comput. Phys. 230 (2011), no. 8, 2952–2976.
  • P. Colella, M. Dorr, J. Hittinger, P. McCorquodale, and D. F. Martin, High-order finite-volume methods on locally-structured grids, Numerical modeling of space plasma flows: ASTRONUM 2008, Astronomical Society of the Pacific Conference Series, no. 406, 2008, pp. 207–216.
  • P. Colella, D. T. Graves, N. D. Keen, T. J. Ligocki, D. F. Martin, P. W. McCorquodale, D. Modiano, P. O. Schwartz, T. D. Sternberg, and B. V. Straalen, Chombo software package for amr applications - design document, 2009.
  • P. Colella and M. D. Sekora, A limiter for PPM that preserves accuracy at smooth extrema, J. Comput. Phys. 227 (2008), no. 15, 7069–7076.
  • P. Colella and P. R. Woodward, The piecewise parabolic method (PPM) for gas dynamical simulations, J. Comput. Phys. 54 (1984), 174–201.
  • P.-W. Fok and R. R. Rosales, Multirate integration of axisymmetric step-flow equations, (2008), submitted to J. Comp. Phys.
  • G. H. Golub and C. F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in Math. Sciences, Johns Hopkins University Press, Baltimore, 1996.
  • C. A. Kennedy and M. H. Carpenter, Additive Runge–Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math. 44 (2003), no. 1-2, 139–181.
  • D. F. Martin, P. Colella, and D. Graves, A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions, J. Comput. Phys. 227 (2008), no. 3, 1863–1886.
  • P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys. 54 (1984), no. 1, 115–173.
  • Q. Zhang, H. Johansen, and P. Colella, A fourth-order accurate finite-volume method with structured adaptive mesh refinement for solving the advection-diffusion equation, preprint, 2010, submitted to SIAM J. Sci. Comp.