Upwinding in the method of lines

Abstract

The method of lines (MOL) is a procedure for the numerical integration of partial differential equations (PDEs). Briefly, the spatial (boundary value) derivatives of the PDEs are approximated algebraically using, for example, finite differences (FDs). If the PDEs have only one initial value variable, typically time, then a system of initial value ordinary differential equations (ODEs) results through the algebraic approximation of the spatial derivatives. If the PDEs are strongly convective (strongly hyperbolic), they can propagate sharp fronts and even discontinuities, which are difficult to resolve in space. Experience has demonstrated that for these systems, some form of upwinding is generally required when replacing the spatial derivatives with algebraic approximations. Here we investigate the performance of various forms of upwinding to provide some guidance in the selection of upwind methods in the MOL solution of strongly convective PDEs.