On monotonicity and boundedness properties of linear multistep methods

Simon Fraser University, Burnaby, British Columbia, V5A 1S6 Canada. Abstract: In this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods. Instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with Runge-Kutta starting procedures. This allows many multistep methods of practical interest to be included in the theory. In a related manner, we also consider contractivity and stability in arbitrary norms

Strong Stability Preserving Two-Step Runge-Kutta Methods

We investigate the strong stability preserving (SSP) property of two-step Runge– Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple\ud subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge–Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations

High-order TVD and TVB linear multistep methods

We consider linear multistep methods that possess the TVD (total variation diminishing) or TVB (total variation bounded) properties, or related general monotonicity and boundedness properties. Strict monotonicity or TVD, in terms of arbitrary starting values for the multistep schemes, is only valid for a small class of methods, under very stringent step size restrictions. This makes them uncompetitive to the TVD Runge-Kutta methods. By relaxing these strict monotonicity requirements a larger class of methods can be considered, including many methods of practical interest. In this paper we construct linear multistep methods of high-order (up to six) that possess relaxed monotonicity or boundedness properties with optimal step size conditions. Numerical experiments show that the new schemes perform much better than the classical TVD multistep schemes. Moreover there is a substantial gain in efficiency compared to recently constructed TVD Runge-Kutta methods

Special boundedness properties in numerical initial value problems

For Runge-Kutta methods, linear multistep methods and other classes of general linear methods much attention has been paid in the literature to important nonlinear stability properties known as total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize conditions guaranteeing these properties were studied by Shu \& Osher (1988) and in numerous subsequent papers. Unfortunately, for many useful methods it has turned out that these properties do not hold. For this reason attention has been paid in the recent literature to the related and more general properties called total-variation-bounded (TVB) and boundedness. In the present paper we focus on stepsize conditions guaranteeing boundedness properties of a special type. These boundedness properties are optimal, and distinguish themselves also from earlier boundedness results by being relevant to sublinear functionals, discrete maximum principles and preservation of nonnegativity. Moreover, the corresponding stepsize conditions are more easily verified in practical situations than the conditions for general boundedness given thus far in the literature. The theoretical results are illustrated by application to the two-step Adams-Bashforth method and a class of two-stage multistep methods

Summation-By-Parts Operators for Time Discretisation: Initial Investigations

Abstract We develop a new high order accurate time-discretisation technique for initial value problems. We focus on problems that originate from a space discretisation using high order finite difference methods on summation-by-parts form with weak boundary conditions, and extend that technique to the time-domain. The new timediscretisation method is global and together with the approximation in space, it generates optimal fully discrete energy estimates, and efficient methods for both stiff and non-stiff problems. In particular, it is shown how stable fully discrete high order accurate approximations of the Maxwells' equations, the elastic wave equations and the linearised Euler and Navier-Stokes equations are obtained. Even though we focus on finite difference approximations, we stress that the methodology is completely general and suitable for all semi-discrete energy-stable approximations

The existence of stepsize-coefficients for boundedness of linear multistep methods

Abstract. This paper deals with general linear multistep methods (LMMs) for the numerical solution of initial value problems. In the context of semi-discretizations of nonlinear time-dependent partial differential equations, much attention was paid to LMMs fulfilling special stability requirements, indicated by the terms total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize restrictions, for the fulfillment of these requirements, were studied by Shu & Osher [J. Comput. Phys., 77 (1988) pp. 439-471 and in numerous subsequent papers. These special stability requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded (TVB). Unfortunately, for many LMMs, the above special requirements are violated, so that one cannot conclude via them that the methods are (totalvariation-)bounded. In this paper, we focus on stepsize restrictions for boundedness directly -rather than via the detour of the above special stability requirements. We present conditions by means of which one can check, for given LMMs, whether or not nontrivial stepsize restrictions exist guaranteeing boundedness. We illustrate the relevance of the above conditions by applying them to various classes of well-known LMMs, hereby supplementing earlier results, for these classes, given in the literature