Total-Variation-Diminishing Time Discretizations

In the computation of conservation laws $u_t + f(u)_x = 0$, total-variation-diminishing (TVD) schemes have been very successful. Many TVD schemes are of method-of-lines form (i.e., discretized in spatial variables only); hence time discretizations that keep TVD and have other properties (e.g., large CFL numbers for steady state calculations, or high-order accuracy for time-dependent problems) are desirable. In this paper we present a class of m-step Runge–Kutta-type TVD time discretizations with large CFL number m, suitable for steady state calculations, and a class of multilevel type TVD high-order time discretizations suitable for time-dependent problems. Some preliminary numerical results are also given.