EXPINT---A MATLAB package for exponential integrators

We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling. For systems of n-component layers and nonidentical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a √ 2:1 length-scale ratio produces an unusual steady square pattern.

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“…(32). For the alternate case, in which the critical modes have a larger wave number than the weakly damped one, Fig.…”

Section: A Length-scale Ratiosmentioning

confidence: 96%

Instabilities and patterns in coupled reaction-diffusion layers

2012

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“…can be expressed in terms of the transfer matrix M (x + D; u, λ) across a period, where M (x; u, λ) is a fundamental solution matrix of the Lax pair (7). Introducing the Floquet discriminant…”

Section: Lax Pair and Characterization Of The Spectrummentioning

confidence: 99%

“…Recent developments [5,11,15,22] have yielded an abundance of possibilities with regards to choice of scheme (see for instance [7] and the accompanying source code for a listing).…”

Section: Exponential Integratorsmentioning

confidence: 99%

“…In order to avoid this in our experiments, these so called ϕ functions are evaluated by scaling the argument z, using a Padé approximant of ϕ , and squaring the result. Details for this are presented in [7] and further analyzed in [8,19]. Stiff order analysis of exponential integrators was introduced as an analytical tool in [15] which addresses the convergence of exponential integrators for semi-linear parabolic partial differential equations.…”

Section: Exponential Integratorsmentioning

confidence: 99%

“…We restrict our attention to explicit exponential integrators to facilitate speed and ease of implementation, and test the two integrators cfree4 and lawson4. The implementation has used the matlab package described in [7] directly. A fourth order split step scheme obtained by Yoshida's technique is included for comparison, as it is, in many ways, related to exponential integrators and it is also used fairly extensively in the physics community.…”

Section: Introductionmentioning

confidence: 99%

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Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

Berland

Islas

Schober

2007

Journal of Computational Physics

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The cubic nonlinear Schrödinger (nls) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The "nonlinear" spectrum of the associated Lax pair reveals topological properties of the nls phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete.

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A block Krylov subspace time‐exact solution method for linear ordinary differential equation systems

Botchev

2013

Numerical Linear Algebra App

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SUMMARYWe propose a time‐exact Krylov‐subspace‐based method for solving linear ordinary differential equation systems of the form y ′ = − Ay + g(t) and y ′ ′ = − Ay + g(t), where y(t) is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term g(t), constructed with the help of the truncated singular value decomposition. The second stage is a special residual‐based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Because both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time‐exact method. Numerical experiments are presented to demonstrate efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

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EXPINT---A MATLAB package for exponential integrators

Cited by 114 publications

References 30 publications

Instabilities and patterns in coupled reaction-diffusion layers

Instabilities and patterns in coupled reaction-diffusion layers

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

A block Krylov subspace time‐exact solution method for linear ordinary differential equation systems

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