Vol. 6, No. 1, 2011

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
An unsplit, higher-order Godunov method using quadratic reconstruction for advection in two dimensions

Sandra May, Andrew Nonaka, Ann Almgren and John Bell

Vol. 6 (2011), No. 1, 27–61
Abstract

Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Important examples include the transport of species and energy in low Mach number models for combustion, atmospheric flows and astrophysics, and contaminant transport in Darcy models of saturated subsurface flow. In this paper, we present a customized finite volume advection scheme for this class of problems that provides accurate resolution for smooth problems while avoiding undershoot and overshoot for nonsmooth profiles. The method is an extension of an algorithm by Bell, Dawson and Shubin (BDS), which was developed for a class of scalar conservation laws arising in porous media flows in two dimensions. The original BDS algorithm is a variant of unsplit, higher-order Godunov methods based on construction of a limited bilinear profile within each computational cell. The new method incorporates quadratic terms in the polynomial reconstruction, thereby reducing the L1 error and better preserving the shape of advected profiles while continuing to satisfy a maximum principle for constant coefficient linear advection. We compare this new method to several other approaches, including the bilinear BDS method and unsplit piecewise parabolic (PPM) methods.

Keywords
Godunov method, scalar conservation law, two-dimensional quadratic reconstruction
Mathematical Subject Classification 2000
Primary: 35-04, 35L65
Milestones
Received: 23 August 2010
Revised: 7 February 2011
Accepted: 19 March 2011
Published: 29 April 2011
Authors
Sandra May
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
Mail Code: 0711
New York, NY 10012
United States
Andrew Nonaka
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
1 Cyclotron Road
MS 50A-1148
Berkeley, CA 94720
United States
Ann Almgren
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
1 Cyclotron Road
MS 50A-1148
Berkeley, CA 94720
United States
John Bell
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
MS 50A-1148
1 Cyclotron Road
Berkeley, CA 94720
United States