Numerical Analysis of a Time-Simultaneous Multigrid Solver for Stabilized Convection-Dominated Transport Problems in 1D

Abstract

The work to be presented focuses on the convection-diffusion equation, especially in the regime of small diffusion coefficients, which is solved using a time-simultaneous multigrid algorithm closely related to multigrid waveform relaxation. For spatial discretization we use linear finite elements, while the time integrator is given by e.g. the Crank-Nicolson scheme. Blocking all time steps into a global linear system of equations and rearranging the degrees of freedom leads to a space-only problem with vector-valued unknowns for each spatial node. Then, common iterative solution techniques, such as the GMRES method with block Jacobi preconditioning, can be used for the numerical solution of the (spatial) problem and allow a higher degree of parallelization in space. We consider a time-simultaneous multigrid algorithm, which exploits space-only coarsening and the solution techniques mentioned above for smoothing purposes. By treating more time steps simultaneously, the dimension of the system of equations increases significantly and, hence, results in a larger number of degrees of freedom per spatial unknown. This can be used to employ parallel processes more efficiently. In numerical studies, the iterative multigrid solution of a problem with up to thousands of blocked time steps is analyzed in 1D. For the special case of the heat equation, it is well known that the number of iterations is bounded above independently of the number of blocked time steps, the time step size, and the spatial resolution. Unfortunately, convergence issues arise for the multigrid solver in convection-dominated regimes. In the context of the standard Galerkin method if the diffusion coefficient is small compared to the grid size and the magnitude of the velocity field, stabilization techniques are typically used to remove artificial oscillations in the solution. However, in our setting, special higher-order variational multiscale-type stabilization methods are discussed, which simultaneously improve the convergence behavior of the iterative solver as well as the smoothness of the numerical solution without significantly perturbing the accuracy.