Monotonicity preserving multigrid time stepping schemes for conservation laws

摘要

In this paper, we propose monotonicity preserving and total variation diminishing (TVD) multigrid methods for solving scalar conservation laws. We generalize the upwind-biased residual restriction and interpolation operators for solving linear wave equations to nonlinear conservation laws. The idea is to define nonlinear restriction and interpolation based on local Riemann solutions. Theoretical analyses have been provided to analyze the monotonicity preserving and TVD properties of the resulting multigrid time stepping schemes. Numerical results are given to verify the theoretical results and demonstrate the effectiveness of the proposed schemes. Two dimensional extension is also discussed.