Implicit-explicit methods for reaction-diffusion problems in pattern formation

Ruuth, Steven J.

doi:10.1007/bf00178771

Cited by 218 publications

(170 citation statements)

References 34 publications

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“…This scheme was successfully used by Madzvamuse [35] to accurately simulate Turing patterns of the Schnakenberg system. To solve the reaction-diffusion system with kinetics (ii) we employed a second order, 3-level, implicit-explicit (IMEX) scheme (2-SBDF) recommended by Ruuth [44] as a good choice for most reaction-diffusion problems for pattern formation, namely:…”

Section: Approximation Of the State And Adjoint Equationsmentioning

confidence: 99%

“…IMEX schemes use an implicit discretization of the diffusion term, and an explicit discretization of the reaction terms. As the scheme 2-SBDF involves three time levels we approximate the solutions at the first time level using a first-order IMEX scheme (1-SBDF) with a small timestep (Ruuth [44]). The numerical schemes used to approximate the linear adjoint equations were similar to the schemes used to approximate the state equations.…”

Section: Approximation Of the State And Adjoint Equationsmentioning

confidence: 99%

“…To circumvent this problem we perturb the stationary states using known functions. For kinetics (i) we choose the initial conditions (see Ruuth [44], or Madzvamuse [35] vðx; 0Þ ¼ 0:937903 þ 0:0016 cosð2pðx þ yÞÞ þ 0:01…”

Section: Construction Of the Target Functionsmentioning

confidence: 99%

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An efficient and robust numerical algorithm for estimating parameters in Turing systems

Garvie

Maini

Trenchea

2010

Journal of Computational Physics

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a b s t r a c tWe present a new algorithm for estimating parameters in reaction-diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer-Meinhardt reaction-diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.

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Section: Approximation Of the State And Adjoint Equationsmentioning

confidence: 99%

Section: Approximation Of the State And Adjoint Equationsmentioning

confidence: 99%

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An efficient and robust numerical algorithm for estimating parameters in Turing systems

Garvie

Maini

Trenchea

2010

Journal of Computational Physics

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“…The iterated semi-implicit method presented here is based on a kind of integrating-factor variation of Crank-Nicolson that makes it L-stable. (An algorithm based on the Modified Crank-Nicolson algorithm [23,24] could be another way to improve the treatment of damping.) Alternatively, one could develop an iterative semi-implicit algorithm starting from BDF2 instead of CN.…”

Section: Robust Dampingmentioning

confidence: 99%

An iterative semi-implicit scheme with robust damping

Loureiro

Hammett

2008

Journal of Computational Physics

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An efficient, iterative semi-implicit (SI) numerical method for the time integration of stiff wave systems is presented. Physics-based assumptions are used to derive a convergent iterative formulation of the SI scheme which enables the monitoring and control of the error introduced by the SI operator. This iteration essentially turns a semi-implicit method into a fully implicit method. Accuracy, rather than stability, determines the timestep. The scheme is secondorder accurate and shown to be equivalent to a simple preconditioning method. We show how the diffusion operators can be handled so as to yield the property of robust damping, i.e., dissipating the solution at all values of the parameter D∆t, where D is a diffusion operator and ∆t the timestep. The overall scheme remains second-order accurate even if the advection and diffusion operators do not commute. In the limit of no physical dissipation, and for a linear test wave problem, the method is shown to be symplectic. The method is tested on the problem of Kinetic Alfvén wave mediated magnetic reconnection. A Fourier (pseudo-spectral) representation is used. A 2-field gyrofluid model is used and an efficacious k-space SI operator for this problem is demonstrated. CPU speed-up factors over a CFL-limited explicit algorithm ranging from ∼ 20 to several hundreds are obtained, while accurately capturing the results of an explicit integration. Possible extension of these results to a real-space (grid) discretization is discussed.

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“…The need for these strategies is by no means limited to Navier-Stokes applications. [7,19,55,69] From a termwise point of view, a linearization of one-dimensional CDR equations can provide insight into the distinguishing characteristics of each term. Upon method-oflines discretization using high-order, finite-difference techniques, the CDR equations may be written as a system of ordinary differential equations (ODES) and analyzed with If the stability domain of the integrator contains all values of zc, zD, and zR, then stable integration can be done.…”

Section: A1 Introductionmentioning

confidence: 99%

Control Strategies for Homogeneous charge compression Ignition Engines: LDRD Final Report

Chen¹

2003

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Implicit-explicit methods for reaction-diffusion problems in pattern formation

Cited by 218 publications

References 34 publications

An efficient and robust numerical algorithm for estimating parameters in Turing systems

An efficient and robust numerical algorithm for estimating parameters in Turing systems

An iterative semi-implicit scheme with robust damping

Control Strategies for Homogeneous charge compression Ignition Engines: LDRD Final Report

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