Optimal multigrid convergence by elliptic/hyperbolic splitting

We describe a multigrid method for solving the steady Euler equations that is optimal in the sense of requiring O(N) operations till convergence, where N is the number of unknowns. The method relies on an elliptic/hyperbolic decomposition achieved by local preconditioning. The splitting allows the embedded advection equations to be treated with streamwise semicoarsening rather than full coarsening, which would not be effective. A simple 2-D numerical computation is presented as proof of concept. A convergence study indicates the split method has complexity O(N) over a wide range of grid spacings and Mach numbers, while the use of full coarsening for all equations makes the complexity deteriorate to almost O(N1.5).