Time-accurate solution of advection-diffusion problems by finite elements

Abstract The Taylor-Galerkin method, recently proposed for the spatial and temporal discretization of hyperbolic equations, is employed to derive accurate and efficient numerical schemes for the solution of time-dependent advection-diffusion problems. Two distinct numerical strategies are discussed: the first is suitable for evolutionary problems, while the second is appropriate for situations in which a steady state is eventually reached. For purely evolutionary problems the Taylor-Galerkin method is applied to the complete advection-diffusion equation and, through modifications of the standard ‘mass’ matrix, is shown to generate an incremental form of the Crank-Nicolson time-stepping method. Such an incremental form also lends itself to the derivation of computationally simple, but accurate, explicit algorithms. To deal with transient situations which evolve towards a highly convective steady state, the ‘global’ Taylor-Galerkin method is found to be ineffective since it reduces to the standard Galerkin formulation as the temporal term vanishes. For such cases we suggest the use of a splitting-up method in which advection and diffusion are treated separately by appropriate Taylor-Galerkin methods. The proposed methods are illustrated first for linear, constant-coefficient equations. They are successively extended to deal with nonlinear and multi-dimensional problems. Numerical applications indicate their effectiveness.