Solution of time-dependent advection-diffusion problems with the sparse-grid combination technique and a Rosenbrock solver

In the current paper the efficiency of the sparse-grid combination technique applied to time-dependent advection-diffusion problems is investigated. For the time integration we employ a third-order Rosenbrock scheme implemented with adaptive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coefficient problem and a system of 2D nonlinear Burgers' equations. In short, the combination technique proved more efficient than a single grid approach for the simpler linear problem. For the Burgers' equations this gain in efficiency was only observed when one of the two solution components was set to zero, making the problem more grid-aligned.