We propose a fast multiplicative and additive multigrid time stepping schemes for solving linear and nonlinear wave equations in one dimension. The idea is based on an upwind biased interpolationand residual restriction operators, and a nonstandard coarse grid update formula for linear equations. We prove that the two-level schemes preserve monotonicity and are total variation diminishing, and the same results hold for the multilevel additive scheme. We generalize the idea to nonlinear equations by solving local Riemann problems. We demonstrate numerically that these schemes are essentially nonoscillatory, and that the optimal speed of wave propagation of 2^M-1 is achieved, where M is the number of grids.

Keywords. monotonicity preserving,total variation diminishing, multigrid time stepping

AMS subject classifications. 65M12, 65M25, 65M55, 65F10