A Space-Time Extrapolation Cascadic Multigrid Method for 2D Linear Parabolic Problems

Abstract

We present a new space-time extrapolation cascadic multigrid method with SSOR preconditioned GMRES smoother for linear parabolic equations. This method simultaneously solves all time steps of the system using an all-at-once approach, employing Crank-Nicolson discretization in time and central difference discretization in space. The key components of the new algorithm are the Richardson extrapolation and Lagrange interpolation operators. By utilizing these techniques with numerical solutions of current and previous grids, we generate a good initial guess for the iterative solution on the next finer grid, greatly reducing the number of required iterations and computational time. Finally, we explain how to implement the new multigrid method and show its efficiency through numerical experiments.