In this paper nonlinear monotonicity and boundedness properties are analyzed for linear multistep methods. We focus on methods which satisfy a weaker boundedness condition than strict monotonicity for arbitrary starting values. In this way, many linear multistep methods of practical interest are included in the theory. Moreover, it will be shown that for such methods monotonicity can still be valid with suitable Runge-Kutta starting procedures. Restrictions on the stepsizes are derived that are not only sufficient but also necessary for these boundedness and monotonicity properties.2000 Mathematics Subject Classification: 65L06, 65M06, 65M20.
Abstract. 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.
Abstract. For Runge-Kutta methods (RKMs), linear multistep methods (LMMs), and classes of general linear methods (GLMs), much attention has been paid, in the literature, to special nonlinear stability requirements indicated by the terms total-variation-diminishing, strong stability preserving, and monotonicity. Stepsize conditions, guaranteeing these properties, were derived 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. Unfortunately, for many well-known methods, the above special requirements are violated, so that one cannot conclude in this way that the methods are (total-variation-)bounded. In this paper, we focus on stepsize conditions for boundedness directly, rather than via the detour of the above special stability properties. We present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual RKMs, LMMs, and GLMs, thereby generalizing results on the special stability properties mentioned above.
MSC:65L05 65L06 65L20 65M20
Keywords:Initial value problem Method of lines (MOL) Multistep methods Monotonicity Boundedness Strong-stability-preserving (SSP) a b s t r a c t One-leg multistep methods have some advantage over linear multistep methods with respect to storage of the past results. In this paper boundedness and monotonicity properties with arbitrary (semi-)norms or convex functionals are analyzed for such multistep methods. The maximal stepsize coefficient for boundedness and monotonicity of a one-leg method is the same as for the associated linear multistep method when arbitrary starting values are considered. It will be shown, however, that combinations of one-leg methods and Runge-Kutta starting procedures may give very different stepsize coefficients for monotonicity than the linear multistep methods with the same starting procedures. Detailed results are presented for explicit two-step methods.
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