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

Spijker, M. N.

doi:10.1016/j.apnum.2012.09.005

Cited by 8 publications

(28 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Common choices for the seminorm · on V in applications include the supremum norm or the total variation seminorm. For LMMs, a more detailed exposition of the above topics together with references can be found, for example, in [16,Section 1]. For Runge-Kutta methods, analogous questions have been analyzed thoroughly and solved satisfactorily in [8].…”

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 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%

“…Since a LMM with larger step-size coefficient is more efficient, one is also interested in the maximum value of the SCM or SCB. Conditions that are easy to check and are necessary and sufficient for the existence of a positive SCM, or for a given positive number to be a SCM have already been devised, see [16,Section 1.1].…”

Section: Introductionmentioning

confidence: 99%

“…The stability region of the LMM, denoted by S, is defined as S := {λ ∈ C : 1 − λb 0 = 0 and ρ − λσ satisfies the root condition}, see [16,Section 2.1]. The interior of the stability region will be denoted by int(S).…”

mentioning

confidence: 99%

See 4 more Smart Citations

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

Lóczi

2017

Numer Algor

View full text Add to dashboard Cite

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).

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

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

Lóczi

2017

Numer Algor

View full text Add to dashboard Cite

show abstract

Optimal subsets in the stability regions of multistep methods

Lóczi

2019

Numer Algor

View full text Add to dashboard Cite

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,

show abstract