Designing optimal smoothers for multigrid methods for unsteady flow problems

We consider the solution of unsteady flow problems using multigrid methods. There, the typical solver comes from a dual time stepping approach, where an existing method designed for a steady problem is used for unsteady flows. This design mismatch leads to a significant decrease in convergence speed. Here, we will use the linear advection equation as a model problem to find optimal Runge-Kutta smoothers specifically for the unsteady case, based on a Fourier analysis. This leads to a significantly improved multigrid method.