An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions
Journal of Computational Physics
Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
An unsplit 3D upwind method for hyperbolic conservation laws
Journal of Computational Physics
High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
On WAF-type schemes for multidimensional hyperbolic conservation laws
Journal of Computational Physics
Wave propagation algorithms for multidimensional hyperbolic systems
Journal of Computational Physics
MPDATA: a finite-difference solver for geophysical flows
Journal of Computational Physics
Journal of Computational Physics
A wave propagation method for three-dimensional hyperbolic conservation laws
Journal of Computational Physics
A conservative three-dimensional Eulerian method for coupled solid-fluid shock capturing
Journal of Computational Physics
Finite volume evolution Galerkin methods for nonlinear hyperbolic systems
Journal of Computational Physics
Adaptive low Mach number simulations of nuclear flame microphysics
Journal of Computational Physics
Finite Volume Evolution Galerkin Methods for Hyperbolic Systems
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Scientific Computing
Finite volume evolution Galerkin (FVEG) methods for three-dimensional wave equation system
Applied Numerical Mathematics
Short Note: A limiter for PPM that preserves accuracy at smooth extrema
Journal of Computational Physics
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Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Of particular interest here is the transport of species and energy in low Mach number models for combustion, atmospheric flows, and astrophysics, as well as contaminant transport in Darcy models of saturated subsurface flow. An important characteristic of these problems is that the velocity field is not known analytically. Instead, an auxiliary equation is solved to compute averages of the velocities over faces in a finite volume discretization. In this paper, we present a customized three-dimensional 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. Here we present a three-dimensional extension of the original BDS algorithm that is based on a limited trilinear profile within each cell. We compare this new method to several other unsplit approaches, including piecewise linear methods, piecewise parabolic methods, and wave propagation schemes.