A numerical framework for singular limits of a class of reaction diffusion problems

Moyles, Iain; Wetton, Brian

doi:10.1016/j.jcp.2015.07.053

Cited by 4 publications

(12 citation statements)

References 35 publications

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“…We will further show for ring-type solutions that there are parameter regimes where there are no O(1) time-scale breakup instabilities associated with the profile of the ring solution, as characterized by the spectrum of a nonlocal eigenvalue problem NLEP. As such, our analysis suggests x-x that there are parameter regimes where the reduced moving-boundary dynamics, which can be computed using the algorithm in [26], should accurately reflect corresponding long-time behavior in the full PDE system (1.1). In the remainder of this manuscript, we will primarily focus on the simple geometrical configuration of a ring-type solution for which the development of an analytical theory is tractable.…”

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 89%

“…While analytic solutions using this formulation are not generally possible for arbitrary curves Γ, recently in the companion article [26], a numerical methodology has been designed to solve moving-boundary problems of the type (2.13). This class of problems is new, as compared to the well-known Cahn-Hilliard problems of material science, in the sense that the normal velocity depends on the average, rather than the difference, of the diffusive flux across the interface [26].…”

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

“…This class of problems is new, as compared to the well-known Cahn-Hilliard problems of material science, in the sense that the normal velocity depends on the average, rather than the difference, of the diffusive flux across the interface [26]. As such, a novel numerical framework was developed in [26] to treat (2.13). Some numerical examples of solutions obtained using this methodology of [26] are presented in Fig.…”

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

“…As such, a novel numerical framework was developed in [26] to treat (2.13). Some numerical examples of solutions obtained using this methodology of [26] are presented in Fig. 2.1, where it is seen that the RD system has a rich set of dynamics to be understood.…”

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

“…(c) r0 = 5 + 0.3 cos(6θ) Figure 2.1: Some numerical solutions to (2.13) using the framework developed in [26]. For (a), we have taken the localized curve to be a near circle with radius given by r 0 = 0.5+0.3 cos(5θ) inside a larger circular domain R = 1.…”

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

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Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

Moyles¹,

Ward²

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. We analyze a singularly perturbed reaction-diffusion system in the semi-strong diffusion regime in two spatial dimensions where an activator species is localized to a closed curve, while the inhibitor species exhibits long range behavior over the domain. In the limit of small activator diffusivity we derive a new moving boundary problem characterizing the slow time evolution of the curve, which is defined in terms of a quasi steady-state inhibitor diffusion field and its properties on the curve. Numerical results from this curve evolution problem are illustrated for the Gierer-Meinhardt model (GMS) with saturation in the activator kinetics. A detailed analysis of the existence, stability, and dynamics of ring and near-ring solutions for the GMS model is given, whereby the activator concentrates on a thin ring concentric within a circular domain. A key new result for this ring geometry is that by including activator saturation there is a qualitative change in the phase portrait of ring equilibria, in that there is an S-shaped bifurcation diagram for ring equilibria, which allows for hysteresis behavior. In contrast, without saturation, it is well-known that there is a saddle-node bifurcation for the ring equilibria. For a near-circular ring, we develop an asymptotic expansion up to quadratic order to fully characterize the normal velocity perturbations from our curve-evolution problem.In addition, we also analyze the linear stability of the ring solution to both breakup instabilities, leading to the disintegration of a ring into localized spots, and zig-zag instabilities, leading to the slow shape deformation of the ring. We show from a nonlocal eigenvalue problem that activator saturation can stabilize breakup patterns that otherwise would be unstable. Through a detailed matched asymptotic analysis, we derive a new explicit formula for the small eigenvalues associated with zig-zag instabilities, and we show that they are equivalent to the velocity perturbations induced by the near-circular ring geometry. Finally, we present full numerical simulations from the GMS PDE system that confirm the predictions of the analysis.

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Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 89%

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

Section: Boundary Fitted Coordinate Formulationmentioning

confidence: 99%

See 3 more Smart Citations

Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

Moyles¹,

Ward²

2017

SIAM J. Appl. Dyn. Syst.

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Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical (PEC) solar cells

Harmon

Gamba

Ren

2016

Journal of Computational Physics

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This work concerns the numerical solution of a coupled system of self-consistent reaction-drift-diffusion-Poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical (PEC) solar cells with reactive semiconductor and electrolyte interfaces. We present three numerical algorithms, mainly based on a mixed finite element and a local discontinuous Galerkin method for spatial discretization, with carefully chosen numerical fluxes, and implicit-explicit time stepping techniques, for solving the time-dependent nonlinear systems of partial differential equations. We perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model.The phenomena of reaction and transport of charged particles in spatial regions with semiconductor and electrolyte interfaces has been extensively investigated in the past few decades [12,49,50,66,74,86,91,94,95,105]; see [75] for recent reviews on the subject. There are three

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Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

Ward

2018

Nonlinearity

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Localized spatial patterns commonly occur for various classes of linear and nonlinear diffusive processes. In particular, localized spot patterns, where the solution concentrates at discrete points in the domain, occur in the nonlinear reactiondiffusion (RD) modeling of diverse phenomena such as chemical patterns, biological morphogenesis, and the spatial distribution of urban crime. In a 2-D spatial domain we survey some recent and new results for the existence, linear stability, and slow dynamics of localized spot patterns by using the Brusselator RD model as the prototypical example. In the context of linear diffusive systems with localized solution behavior, we will discuss some previous results for the determination of the mean first capture time for a Brownian particle in a 2-D domain with localized traps, and the determination of the persistence threshold of a species in a 2-D landscape with patchy food resources. Common features in the analysis of all of these spatially localized patterns are emphasized, including the key role of certain matrices involving various Green's functions, and the derivation and study of new classes of interacting particle systems and discrete variational problems arising from various asymptotic reductions. The mathematical tools include matched asymptotic analysis based on strong localized perturbation theory, spectral analysis, the analysis of nonlocal eigenvalue problems, and bifurcation theory. Some specific open problems are highlighted and, more broadly, we will discuss a few new research frontiers for the analysis of localized patterns in multi-dimensional domains.

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A numerical framework for singular limits of a class of reaction diffusion problems

Cited by 4 publications

References 35 publications

Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical (PEC) solar cells

Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

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