An incremental multigrid strategy for the fluid dynamics equations

This paper provides a novel incremental multigrid strategy for the equations of fluid dynamics. The (time dependent) governing equations are discretized in time by means of a two level implicit Euler scheme and linearized using Taylor series and the incremental (delta) form of Beam and Warming. The coefficients and the right hand side of the resulting linear systems are evaluated always at the finest grid level, whereas the (delta) unknowns are computed (approximately, by a single relaxation sweep) on a sequence of coarser meshes. At every grid level the computed deltas are interpolated up to the finest-grid level and used to update the solution, as well as the coefficients and the right hand side of the linear systems. This process is repeated, sweeping all grid levels successively, until a satisfactory convergence criterion is met. The validity of the proposed approach is demonstrated by solving a simple linear problem and the vorticity-stream function Navier-Stokes equations, using line relaxation methods as smoothers, and the lambda-formulation Euler equations, in conjunction with a simple explicit smoother. In all cases, the proposed multigrid strategy provides a considerable efficiency gain over the corresponding single-grid methods.