Contractivity-Preserving Implicit Linear Multistep Methods

Lenferink, H. W. J.

doi:10.2307/2008536

Cited by 11 publications

(18 citation statements)

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“…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%

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: 99%

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

2011

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“…In [14], Hundsdorfer, Ruuth and Spiteri explain that it follows from Lenferink's results on contractivity for linear systems [19] that, in general, any two step method of order p > 1 would have CFL coefficient no greater than c = 2. Of course, this provides a bound on the results for nonlinear problems as well.…”

Section: Implicit Multi-step Methodsmentioning

confidence: 99%

On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

2005

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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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“…The present algorithm was also applied to find such optimal polynomials, in [14]. Optimal linear multistep methods were considered in [19,20,10,8]), and are investigated further in Section 3 of the present work.…”

Section: The Feasibility Problemmentioning

confidence: 99%

“…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%

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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 Implicit Linear Multistep Methods

Cited by 11 publications

References 1 publication

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

Computation of optimal monotonicity preserving general linear methods

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