Sources and Multidimensions

This chapter addresses two major issues when solving advection-reaction partial differential equations (PDEs) in general domains, namely the presence of source terms in the PDEs and the multidimensional character of the computational domain. We present simple but effective methods to deal with both situations. First, 1D advection-reaction systems are split into an advection system of PDEs and a reaction part consisting of a system of Ordinary Differential Equations (ODEs). We call the procedure advection-reaction splitting. For the advection part, there are several schemes presented in previous chapters that are directly applicable. For the reaction part, a section on numerical methods for ODEs is included. First and second-order advection-reaction splitting schemes are formulated. For multidimensional advection systems, we present two approaches. The first one relies on dimensional splitting, whereby 1D schemes applied to augmented 1D advection systems are solved in succession. First and second-order dimensional splitting schemes are presented. The second approach rests on unsplit, simultaneous-update finite volume formulations, whereby the updating includes contributions from all element-edge fluxes. This approach is suitable for unstructured meshes. A major issue enters the scene in this second class of methods, namely stability. A section on stability analysis for multidimensional schemes is included, and examples of stable schemes are given. Finally, we present a particular class of unsplit second-order finite volume methods of the WAF type, which exhibit very generous stability regions in 2D and 3D. Moreover, the WAF framework may be extended to other classes of schemes, including centred schemes.