Numerical Study of SUPG and LPS Methods Combined with Higher Order Variational Time Discretization Schemes Applied to Time-Dependent Linear Convection-Diffusion-Reaction Equations

This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamlineupwind Petrov–Galerkin (SUPG) and local projection stabilization (LPS) meth-ods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) meth-ods and continuous Galerkin–Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. In addition, the long-time behavior of overshoots and undershoots is studied. The efficient solution of the obtained systems of linear equations by GMRES methods with multigrid preconditioners will be investigated.