Advection Problems with Spatially Varying Velocity Fields: 1D and 2D Analytical and Numerical Solutions

Abstract

The advection equation is one of the most commonly used models in environmental fluid dynamics. Despite this, there exist very few analytical solutions to this equation for the case where the velocity field is spatially varying. In this work a class of solutions to the conservation form of the advection equation is defined in one and two dimensions using a particular change of variables. Example solutions are developed for constant, varying, and discontinuous initial density profiles, as well as for continuous and discontinuous velocity fields. The accuracy of four common finite-volume numerical methods is evaluated for eight test cases and compared to the exact solutions. The first- and second-order upwind methods and the upwind method with the van Leer and Superbee slope limiters are used. It is found that the upwind method with either of the slope limiters is well suited to numerically solving the advection equation with spatially varying velocity fields for most test cases, while, as is the case for nondivergent velocity fields, the first- and second-order upwind methods have some serious drawbacks.