Stepsize Conditions for General Monotonicity in Numerical Initial Value Problems

For Runge–Kutta methods and linear multistep methods, much attention has been paid, in the literature, to special nonlinear stability properties indicated by the terms total-variationdiminishing (TVD), strong-stability-preserving (SSP), and monotonicity. Stepsize conditions, guaranteeing these properties, were studied, e.g., by Shu and Osher [J. Comput. Phys., 77 (1988), pp. 439–471], Gottlieb, Shu, and Tadmor [SIAM Rev., 43 (2001), pp. 89–112], Hundsdorfer and Ruuth [Monotonicity for Time Discretizations, Dundee Conference Report NA/217 2003, University of Dundee, Dundee, UK, 2003, pp. 85–94], Higueras [J. Sci. Comput., 21 (2004), pp. 193–223] and [SIAM J. Numer. Anal., 43 (2005), pp. 924–948], Spiteri and Ruuth [SIAM J. Numer. Anal., 40 (2002), pp. 469–491], Gottlieb [J. Sci. Comput., 25 (2005), pp. 105–128], and Ferracina and Spijker [SIAM J. Numer. Anal., 42 (2004), pp. 1073–1093] and [Math. Comp., 74 (2005), pp. 201–219]. In the present paper, we obtain a special stepsize condition guaranteeing the above properties, for a generic numerical process. This condition is best possible in a well defined and natural sense. It is applicable to the important class of general linear methods, and it can also be used to answer some open questions, for methods of which the above stability properties were studied earlier.