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Finite Difference WENO Schemes with Lax-Wendroff-Type Time Discretizations

In this paper we develop a Lax--Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge--Kutta time discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax--Wendroff time discretization procedure is more cost effective than the Runge--Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.