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-stability-preserving (SSP) property (see, e.g., Gottlieb, Shu, and ...

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