Monotonicity-Preserving Linear Multistep Methods

Hundsdorfer, Willem; Ruuth, Steven J.; Spiteri, Raymond J.

doi:10.1137/s0036142902406326

Cited by 86 publications

(92 citation statements)

References 19 publications

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“…e.g. Ferracina & Spijker [4], Gottlieb, Ketcheson & Shu [5], Gottlieb, Shu, & Tadmor [6], Higueras [9,10], Hundsdorfer & Ruuth [14,15], Hundsdorfer, Ruuth & Spiteri [16], Shu & Osher [20], Spijker [22]. In most papers, the focus has been on the situation where (1.2) stands for just one step (N = 1) of a GLM and (1.5)…”

Section: Bounds For Numerical Approximationsmentioning

confidence: 99%

Special boundedness properties in numerical initial value problems

2011

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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.

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Section: Bounds For Numerical Approximationsmentioning

confidence: 99%

Special boundedness properties in numerical initial value problems

2011

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“…is unconditionally strongly stable, u n+1 u n [14]. If so, then (4.1) would be unconditionally strongly stable under the same norm provided β i > 0 for all i.…”

Section: Diagonally Implicit Runge-kutta Methodsmentioning

confidence: 99%

“…An illustration of of spatial discretizations which possess strong stability properties for implicit Euler. In fact, it was shown in [12,14], that any spatial discretization L, which is strongly stable in some norm for the explicit forward Euler method under a certain time restriction will also be strongly stable, in the same norm, for the implicit Euler method, without a time restriction.…”

Section: Multi Step Methodsmentioning

confidence: 99%

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On high order strong stability preserving runge-kutta and multi step time discretizations

Gottlieb

2005

J Sci Comput

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Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm-of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.

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“…different from . T V , and also when solving differential equations different from conservation laws (see, e.g., Dekker and Verwer [3], Hundsdorfer and Verwer [13], LeVeque [17] [7], Hundsdorfer, Ruuth and Spiteri [12], Morton [18]). …”

Section: A Generalization Of the Shu-osher Process (18)mentioning

confidence: 99%

An extension and analysis of the Shu-Osher representation of Runge-Kutta methods

Ferracina

Spijker

2004

Math. Comp.

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Abstract. In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is totalvariation-diminishing (TVD). Much attention has been paid to these representations in the literature.In general, a Shu-Osher representation of a given Runge-Kutta method is not unique. Therefore, of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them.Several basic questions are still open, notably regarding the following issues: (1) the formulation of a simple and general strategy for finding a best possible Shu-Osher representation for any given Runge-Kutta method; (2) the question of whether the TVD property of a given Runge-Kutta method can still be guaranteed when the stepsize condition, corresponding to a best possible ShuOsher representation of the method, is violated; (3) the generalization of the Shu-Osher approach to general (possibly implicit) Runge-Kutta methods.In this paper we give an extension and analysis of the original Shu-Osher representation, by means of which the above questions can be settled. Moreover, we clarify analogous questions regarding properties which are referred to, in the literature, by the terms monotonicity and strong-stability-preserving (SSP).

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Monotonicity-Preserving Linear Multistep Methods

Abstract: C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Modelling, Analysis and SimulationMonotonicity-preserving linear multistep methods

Cited by 86 publications

References 19 publications

Special boundedness properties in numerical initial value problems

Special boundedness properties in numerical initial value problems

On high order strong stability preserving runge-kutta and multi step time discretizations

An extension and analysis of the Shu-Osher representation of Runge-Kutta methods

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