Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

Hundsdorfer, Willem; Mozartova, Anna; Spijker, M. N.

doi:10.1007/s10915-011-9487-1

Cited by 15 publications

(27 citation statements)

References 26 publications

(46 reference statements)

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“…These investigations confirm and extend some earlier results [5,6,7,9] concerning the existence of step-size coefficients for monotonicity or step-size coefficients for boundedness. The results of [16,Section 5] have the following form.…”

Section: The Existence Of a Scbsupporting

confidence: 90%

“…The theoretical framework we use is presented in [9,16], while the computational techniques we apply show many similarities with those of [10]. All computations in this work have been performed by using Mathematica 10.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 1.1 we present some definitions and notation. In Sections 1.2 and 1.3 we review the main results of [9] and [16] concerning (ii ) and (i ) above, respectively. Section 2 contains our theorems for three families of multistep methods:…”

Section: Introductionmentioning

confidence: 99%

“…The above theorem is based on the material developed in [9]. In [9, Section 6], the authors numerically determine the maximum SCB values for members of several parametric families of LMMs by repeatedly applying the following test.…”

mentioning

confidence: 99%

“…For a particular LMM and given γ > 0, they check if γ is a SCB by choosing a large N ∈ N, and verifying µ n (γ) ≥ 0 for all 1 ≤ n ≤ N . However, as the authors point out in [9], it is not obvious (neither a priori nor a posteriori ) how large N one should choose to conclude-with high certainty-that µ n (γ) ≥ 0 for all n ∈ N + . They typically use N ≈ 10 3 ; as a comparison, see our Remark 3.4.…”

mentioning

confidence: 99%

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Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Lóczi

2017

Numer Algor

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Linear multistep methods (LMMs) applied to approximate the solution of initial value problems-typically arising from method-of-lines semidiscretizations of partial differential equations-are often required to have certain monotonicity or boundedness properties (e.g. strong-stability-preserving, total-variation-diminishing or totalvariation-boundedness properties). These properties can be guaranteed by imposing step-size restrictions on the methods. To qualitatively describe the step-size restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) or its generalization, the step-size coefficient for boundedness (SCB). A LMM with larger SCM or SCB is more efficient, and the computation of the maximum SCM for a particular LMM is now straightforward. However, it is more challenging to decide whether a positive SCB exists, or determine if a given positive number is a SCB. Theorems involving sign conditions on certain linear recursions associated to the LMM have been proposed in the literature that allow us to answer the above questions: the difficulty with these theorems is that there are in general infinitely many sign conditions to be verified. In this work we present methods to rigorously check the sign conditions. As an illustration, we confirm some recent numerical investigations concerning the existence of positive SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a stronger result, we determine the optimal values of the SCBs as exact algebraic numbers in the BDF family (with 1 ≤ k ≤ 6 steps) and in the Adams-Bashforth family (with 1 ≤ k ≤ 3 steps).

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Section: The Existence Of a Scbsupporting

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Lóczi

2017

Numer Algor

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Optimal subsets in the stability regions of multistep methods

Lóczi

2019

Numer Algor

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In this work we study the stability regions of linear multistep or multiderivative multistep methods for initial-value problems by using techniques that are straightforward to implement in modern computer algebra systems. In many applications, one is interested in (i ) checking whether a given subset of the complex plane (e.g. a sector, disk, or parabola) is included in the stability region of the numerical method, (ii ) finding the largest subset of a certain shape contained in the stability region of a given method, or (iii ) finding the numerical method in a parametric family of multistep methods whose stability region contains the largest subset of a given shape.First we describe a simple procedure to exactly calculate the stability angle α in the definition of A(α)-stability by representing the root locus curve (RLC) of the multistep method as an implicit algebraic curve. As an illustration, we consider two finite families of implicit multistep methods. We exactly compute the stability angles for the k-step BDF methods (3 ≤ k ≤ 6) and discover that the values of tan(α) are surprisingly simple algebraic numbers of degree 2, 2, 4 and 2, respectively. In contrast, the corresponding values of tan(α) for the k-step second-derivative multistep methods of Enright (3 ≤ k ≤ 7) are much more complicated; the smallest algebraic degree here is 22.Next we determine the exact value of the stability radius in the BDF family for each 3 ≤ k ≤ 6, that is, the radius of the largest disk in the left half of the complex plane, symmetric with respect to the real axis, touching the imaginary axis and lying in the stability region of the corresponding method. These radii turn out to be algebraic numbers of degree 2, 3, 5 and 5, respectively.Finally, we demonstrate how some Schur-Cohn-type theorems of recursive nature and not relying on the RLC method can be used to exactly solve some optimization problems within infinite parametric families of multistep methods. As an example, we choose a two-parameter family of implicit-explicit (IMEX) methods: we identify the unique method having the largest stability angle in the family, then we find the unique method in the same family whose stability region contains the largest parabola. * LLoczi@cs.elte.hu,

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The existence of stepsize-coefficients for boundedness of linear multistep methods

Spijker

2013

Applied Numerical Mathematics

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Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

Cited by 15 publications

References 26 publications

Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Optimal subsets in the stability regions of multistep methods

The existence of stepsize-coefficients for boundedness of linear multistep methods

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