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

Abstract

This work focuses on the solution of the convection–diffusion equation, especially for small diffusion coefficients, employing a time-simultaneous multigrid algorithm, which is closely related to multigrid waveform relaxation. For discretization purposes, linear finite elements are used while the Crank–Nicolson scheme acts as the time integrator. By combining all time steps into a global linear system of equations and rearranging the degrees of freedom, a space-only problem is formed with vector-valued unknowns for each spatial node. The generalized minimal residual method with block Jacobi preconditioning can be used to numerically solve the (spatial) problem, allowing a higher degree of parallelization in space. A time-simultaneous multigrid approach is applied, utilizing space-only coarsening and the aforementioned solution techniques for smoothing purposes. Numerical studies analyze the iterative solution technique for 1D test problems. For the heat equation, the number of iterations stays bounded independently of the number of time steps, the time increment, and the spatial resolution. However, convergence issues arise in situations where the diffusion coefficient is small compared to the grid size and the magnitude of the velocity field. Therefore, a higher-order variational multiscale stabilization is used to improve the convergence behavior and solution smoothness without compromising its accuracy in convection-dominated scenarios.