TOTAL VARIATION DIMINISHING RUNGE-KUTTA SCHEMES

In this paper we further explore a class of high order TVD (total vari ation diminishing) Runge-Kutta time discretization initialized in a pa per by Shu and Osher, suitable for solving hyperbolic conservation law s with stable spatial discretizations. We illustrate with numerical ex amples that non-TVD but linearly stable Runge-Kutta time discretizatio n can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta meth ods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods ibr second, third and fourth order, an d for low storage Runge-Kutta methods.