Optimizing a multigrid Runge-Kutta smoother for variable-coefficient convection-diffusion equations

The theory of Generally Locally Toeplitz (or GLT for short) sequences of matrices is proposed in the analysis of a multigrid solver for the linear systems generated by finite volume/finite difference approximations of variable-coefficients linear convection-diffusion equations in 1D, proposed by Birken in 2012, and extended here to 2D problems. The multigrid solver is used with a Runge-Kutta smoother. Optimal coefficients for the smoother are found by considering the unsteady linear advection equation and using optimization algorithms. In particular, in order to reduce the issues of having multiple local minima, the sequential quadratic programming (SQP) mixed with genetic and particle swarm optimization algorithms are proposed. Numerical results show that our proposals are competitive with respect to other multigrid implementations.