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 $L^1$ 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.

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Mathematical Subject Classification 2000

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