Strong stability of singly-diagonally-implicit Runge--Kutta methods

This paper deals with the numerical solution of initial value problems, for systems of ordinary differential equations, by Runge-Kutta methods (RKMs) with special nonlinear stability properties indicated by the terms total-variation-diminishing (TVD), strongly stable and monotonic. Stepsize conditions, guaranteeing these properties, were studied earlier, see e.g. Shu and Osher [C.-W. Shu, S. Osher, J. Comput. Phys. 77 (1988) 439-471], Gottlieb et al. [S. Gottlieb, C.-W. Shu, E. Tadmor, SIAM Rev. 43 (2001) 89-112], Hundsdorfer and Ruuth [W.H. Hundsdorfer, S.J. Ruuth, Monotonicity for time discretizations, in: D.F. Griffiths, G.A. Watson (Eds.), Proc. Dundee Conference 2003, Report NA/217, Univ. Dundee, 2003, pp. 85-94], Higueras [I. Higueras, J. Sci. Computing 21 (2004) 193-223; I. Higueras, SIAM J. Numer. Anal. 43 (2005) 924-948], Gottlieb [S. Gottlieb, J. Sci. Computing 25 (2005) 105-128], Ferracina and Spijker [L. Ferracina, M.N. Spijker, SIAM J. Numer. Anal. 42 (2004) 1073-1093; L. Ferracina, M.N. Spijker, Math. Comp. 74 (2005) 201-219]. Special attention was paid to RKMs which are optimal, in that the corresponding stepsize conditions are as little restrictive as possible within a given class of methods. Extensive searches for such optimal methods were made in classes of explicit RKMs, see e.g. Gottlieb and Shu [S. Gottlieb, C.-W. Shu, Math. Comp. 67 (1998) 73-85], Spiteri and Ruuth [R.J. Spiteri, S.J. Ruuth, SIAM J. Numer. Anal. 40 (2002) 469-491; R.J. Spiteri, S.J. Ruuth, Math. Comput. Simulation 62 (2003) 125-135], Ruuth [S.J. Ruuth, Math. Comp. 75 (2006) 183-207]. In the present paper we search for methods that are optimal in the above sense, within the interesting class of singly-diagonally-implicit Runge-Kutta (SDIRK) methods, with s stages and order p. Some methods, with 1==3) which we conjecture to be optimal for p=2 and p=3, respectively. Furthermore we prove, for strongly stable SDIRK methods, the order barrier p=<4. We perform numerical experiments, to compare the theoretical properties of various optimal SDIRK methods to the actual TVD properties in the solution of a nonlinear test equation, the 1-dimensional Buckley-Leverett equation.

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