Stepsize Restrictions for the Total-Variation-Diminishing Property in General Runge-Kutta Methods

Much attention has been paid in the literature to total-variation-diminishing (TVD) numerical processes in the solution of nonlinear hyperbolic differential equations. For special Runge– Kutta methods, conditions on the stepsize were derived that are sufficient for the TVD property; see, e.g., Shu and Osher [J. Comput. Phys., 77 (1988), pp. 439–471] and Gottlieb and Shu [Math. Comp., 67 (1998), pp. 73–85]. Various basic questions are still open regarding the following issues: 1. the extension of the above conditions to more general Runge–Kutta methods; 2. simple restrictions on the stepsize which are not only sufficient but at the same time necessary for the TVD property; and 3. the determination of optimal Runge–Kutta methods with the TVD property. In this paper we propose a theory by means of which we are able to clarify the above questions. Moreover, by applying our theory, we settle analogous questions regarding the related strong-stabilitypreserving (SSP) property (see, e.g., Gottlieb, Shu, and Tadmor [SIAM Rev., 43 (2001), pp. 89–112] and Shu [Collected Lectures on the Preservation of Stability under Discretization, D. Estep and S. Tavener, eds., SIAM, Philadelphia, 2002]). Our theory can be viewed as a variant to a theory of Kraaijevanger [BIT, 31 (1991), pp. 482–528] on the contractivity of Runge–Kutta methods.