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

Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations

We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are also studied. In the case of the Boltzmann operator, the methods are based on the introduction of a penalization technique for the collision integral. This reformulation of the collision operator permits to construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator. Finally we show some numerical results which confirm the theoretical analysis

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

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

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