Multigrid stabilization of time integrators for nonlinear parabolic PDEs

We propose to enhance the stability of IMEX schemes applied to the numerical solution of nonlinear parabolic equations by using proper smoothing procedures on bands of frequency components of the solution. The separation of those different components is realized by considering a multigrid (MG) approach. The time scheme on the coarsest grid is implicit and energy diminishing, while time marching is done by a weighted damped IMEX scheme on the finer grids. This approach allows to concentrate most computational efforts on a small set of components (the coarsest grid) and to fine-tune the damping parameters so that there is limited consistency perturbation while enhancing stability. We provide stability results and numerical illustrations on pattern dynamics, Swift-Hohenberg, and crystal-phase fields equations, in both 2D and 3D cases. The results obtained with MG are comparable with those given by a one-grid simulation.