A finite volume upwind scheme for the solution of the linear advection–diffusion equation with sharp gradients in multiple dimensions

A finite volume upwind numerical scheme for the solution of the linear advection equation in multiple dimensions on Cartesian grids is presented. The small-stencil, Modified Discontinuous Profile Method (MDPM) uses a sub-cell piecewise constant reconstruction and additional information at the cell interfaces, rather than a spatial extension of the stencil as in usual methods. This paper presents the MDPM profile reconstruction method in one dimension and its generalization and algorithm to two- and three-dimensional problems. The method is extended to the advection–diffusion equation in multiple dimensions. The MDPM is tested against the MUSCL scheme on two- and three-dimensional test cases. It is shown to give high-quality results for sharp gradients problems, although some scattering appears. For smooth gradients, extreme values are best preserved with the MDPM than with the MUSCL scheme, while the MDPM does not maintain the smoothness of the original shape as well as the MUSCL scheme. However the MDPM is proved to be more efficient on coarse grids in terms of error and CPU time, while on fine grids the MUSCL scheme provides a better accuracy at a lower CPU.