Numerical Evaluation of the Evans Function by Magnus Integration

Aparicio, N.D.; Malham, Simon J. A.; Oliver, Marcel

doi:10.1007/s10543-005-0001-8

Cited by 21 publications

(27 citation statements)

References 32 publications

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“…However, for other equations the wedge (47) and the regime in which our analysis is valid may overlap. We mentioned in the introduction that this work was instigated by the paper of Aparicio, Malham and Oliver [3]. That paper studies a similar equation in the regime arg λ = π, while the analysis in the current paper concerns the complementary regime | arg λ| < π − ε with ε > 0.…”

Section: Discussionmentioning

confidence: 93%

“…The analysis is corroborated by numerical experiments in Section 5. In the final section, we compare our results with those of Aparicio, Malham and Oliver [3] and we discuss the stability of the Magnus method in general. More details of the intricate calculations presented in Sections 2-4 can be found in the technical report [29].…”

Section: Introductionmentioning

confidence: 86%

“…The Evans function near infinity. We are interested in the behaviour of D(λ) and numerical approximations to D(λ) as |λ| → ∞, because experiments show an unexpected deterioration of the approximations in this limit [3]. For simplicity, we will restrict ourselves to scalar reaction-diffusion equations, i.e., we assume that n = 1.…”

Section: The Evans Functionmentioning

confidence: 99%

“…This function was called the Evans function by Alexander, Gardner and Jones [2], who generalized its definition considerably. Since then, the Evans function has been used frequently for stability analysis; see, for example, [1,3,4,6,15] for numerical computations employing the Evans function and [11,22,30,34,35] for an analytic treatment. The review paper by Sandstede [33] gives an excellent overview of the field.…”

Section: Introductionmentioning

confidence: 99%

“…Afendikov and Bridges [1] noticed that when λ grows, the problem may become stiff, and therefore they use a Gauss-Legendre method. Recently, Aparicio, Malham and Oliver [3] proposed a new procedure based on the Magnus expansion, building on the work of Moan [25] and Greenberg and Marletta [14] who used the Magnus expansion to solve Sturm-Liouville problems. Aparicio et al noticed that the Magnus method suffers from order reduction in the stiff regime.…”

Section: Introductionmentioning

confidence: 99%

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Evaluating the Evans function: Order reduction in numerical methods

Malham

Niesen

2008

Math. Comp.

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Abstract. We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss-Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schrödinger operator. We show that although the fourth-order Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourth-order convergence in the end result is preserved. The Gauss-Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.

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Section: Discussionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 86%

Section: The Evans Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Evaluating the Evans function: Order reduction in numerical methods

Malham

Niesen

2008

Math. Comp.

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The Evans Function for nth-Order Operators on the Real Line

Kapitula

Promislow

2013

Applied Mathematical Sciences

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Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals

Beyn

Latushkin

Rottmann-Matthes

2013

Integr. Equ. Oper. Theory

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Abstract. Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains.In this paper we propose to detect and approximate the point spectra of such operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh' theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are well-controlled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schrödinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHugh-Nagumo system.We also relate the contour method to the well-known Evans function and show that our approach provides an alternative to evaluating and computing its zeroes.

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Numerical Evaluation of the Evans Function by Magnus Integration

Cited by 21 publications

References 32 publications

Evaluating the Evans function: Order reduction in numerical methods

Evaluating the Evans function: Order reduction in numerical methods

The Evans Function for nth-Order Operators on the Real Line

Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals

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