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
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.