Contractivity preserving explicit linear multistep methods

Lenferink, H. W. J.

doi:10.1007/bf01406515

Cited by 38 publications

(52 citation statements)

References 11 publications

(15 reference statements)

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“…For instance, as shown in [16], for an explicit k-step method (k > 1) of order p we have c ≤ (k − p)/(k − 1). Most explicit methods used in practice have p = k, and for such methods we cannot have c > 0.…”

Section: Monotonicity and Boundedness For Linear Multistep Methodsmentioning

confidence: 98%

“…More recently, with arbitrary seminorms or more general convex functionals, the term SSP (strong stability preserving) -introduced in [5] -has become popular. Related work for nonlinear problems was done in [16,17,20,24] for contractivity, where one considers ũ n − u n with differences of two numerical solutions instead of u n as in (1.7). Finally we mention that related results on nonnegativity preservation and contractivity or monotonicity for linear problems were derived already in [1,22], again for methods with all a j , b j ≥ 0 and with ∆t ≤ c τ 0 .…”

Section: Monotonicity and Boundedness For Linear Multistep Methodsmentioning

confidence: 99%

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Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

2011

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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.

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Section: Monotonicity and Boundedness For Linear Multistep Methodsmentioning

confidence: 98%

Section: Monotonicity and Boundedness For Linear Multistep Methodsmentioning

confidence: 99%

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

2011

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“…[80, p. 123]. The absolute monotonicity radius R (A,β,γ) and its computation are analyzed, e.g., in [59,61,68,87,129] and [100,101,84] and the references therein, where R (A,β,γ) is discussed in the context of contractivity preserving one-and multistep methods.…”

Section: Characterization Of Other Flow Propertiesmentioning

confidence: 99%

“…[20,19,18,45,47,80,83,113,114,126] and the references therein. Numerical methods that preserve the property of positivity within the discretization have been discussed in [13,76,77,78,80] and in the context of stability, contractivity or monotonicity preserving methods in [59,61,68,84,87,100,101,129]. Due to balance equations, conservation laws or limitation of resources, these processes often are subject to additional constraints leading to the notion of positive DAEs.…”

Section: Introductionmentioning

confidence: 99%

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

Baum

2015

J Dyn Diff Equat

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Differential-algebraic equations (DAEs) are coupled systems of differential and algebraic equations arising in the modeling of dynamic processes that are restricted by auxiliary, algebraic constraints. Examples are, e.g., multibody systems, where the movement of the system is forced to a predefined trajectory by connected joints, electrical circuits or networks, where connections and loops lead to algebraic relations of the flow and effort variables, or biological systems, reaction networks or advection-diffusion equations, where the evolution of the process is restricted by balance equations and conservation laws. Modeling dynamic processes in economic or social sciences, in biological or chemical engineering, the analyzed values typically represent real-valued quantities like the amount of goods or individuals or the density of a chemical or biological species that cannot take negative values. In combination with the algebraic constraints, this property leads to the notion of positive DAEs, i.e., systems whose solutions remain componentwise nonnegative whenever the initial value is nonnegative.In this work, we generalize the concept of the flow from ordinary differential equations (ODEs) to DAEs and use this framework to study system properties like invariant sets and positivity. To decouple the differential and algebraic components in a DAE, we develop an approach that allows to parameterize embedded submanifolds using projections and that extends the idea of projections to embedded submanifold. Combined with the theory of the strangeness-index, we apply this projection approach to remodel a given DAE as a set of explicit differential and algebraic equations. Solving the decoupled system and showing existence and uniqueness of the solution representation, we generalize the notion of the flow to DAEs. For linear systems, we generalize Duhamel's formula. Based on the flow, we extend the results of invariant sets from ODEs to DAEs and we characterize linear and nonlinear DAEs with regard to positivity. ZUSAMMENFASSUNG

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“…Several authors have considered the monotonicity property (4) in the simpler case that the function F in (1) is linear and autonomous [25,17,26,19,20,5]. Then (1) simplifies to…”

Section: Introductionmentioning

confidence: 99%

Computation of optimal monotonicity preserving general linear methods

Ketcheson

2009

Math. Comp.

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Abstract. Monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution. We formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations, and propose an efficient algorithm for its solution. This algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and/or stages. The optimality of some recently proposed methods is verified, and many more efficient methods are found. We use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type, including methods for hyperbolic PDEs that use downwind-biased operators.

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Contractivity preserving explicit linear multistep methods

Cited by 38 publications

References 11 publications

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

Computation of optimal monotonicity preserving general linear methods

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