In this paper we study the large time step (LTS) Godunov scheme for scalar hyperbolic conservation laws proposed by LeVeque. We show that for an arbitrary Courant number, all the possible wave interactions in each time step occur only in a finite number of cells, and the number of cells is bounded by a constant depending on the Courant number for a given initial value problem. As an application of the result mentioned above, we show that for any given Courant number, if the total variation of the initial value satisfies some conditions, then the numerical solutions of the LTS Godunov scheme will converge to the entropy solutions of the convex scalar conservation laws.
"On large time step Godunov scheme for hyperbolic conservation laws." Commun. Math. Sci. 2 (3) 477 - 495, September 2004.