Explicitly Solvable Nonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction‐Diffusion Systems

The transverse stability of localized stripe patterns for certain singularly perturbed two‐component reaction‐diffusion (RD) systems in the asymptotic limit of a large diffusivity ratio is analyzed. In this semi‐strong interaction regime, the cross‐sectional profile of the stripe is well‐approximated by a homoclinic pulse solution of the corresponding 1‐D problem. The linear instability of such homoclinic stripes to transverse perturbations is well known from numerical simulations to be a key mechanism for the creation of localized spot patterns. However, in general, owing to the difficulty in analyzing the associated nonlocal and nonself‐adjoint spectral problem governing stripe stability for these systems, it has not previously been possible to provide an explicit analytical characterization of these instabilities, including determining the growth rate and the most unstable mode within the band of unstable transverse wave numbers. Our focus is to show that such an explicit characterization of the transverse instability of a homoclinic stripe is possible for a subclass of RD system for which the analysis of the underlying spectral problem reduces to the study of a rather simple algebraic equation in the eigenvalue parameter. Although our simplified theory for stripe stability can be applied to a class of RD system, it is illustrated only for homoclinic stripe and ring solutions for a subclass of the Gierer–Meinhardt model and for a three‐component RD system modeling patterns of criminal activity in urban crime.     

The Existence and Stability of Spike Equilibria in the One-Dimensional Gray–Scott Model: The Low Feed-Rate Regime

In a singularly perturbed limit of small diffusivity ɛ of one of the two chemical species, equilibrium spike solutions to the Gray–Scott (GS) model on a bounded one-dimensional domain are constructed asymptotically using the method of matched asymptotic expansions. The equilibria that are constructed are symmetric k -spike patterns where the spikes have equal heights. Two distinguished limits in terms of a dimensionless parameter in the reaction-diffusion system are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime, the solution branches of k -spike equilibria are found to have a saddle-node bifurcation structure. The stability properties of these branches of solutions are analyzed with respect to the large eigenvalues λ in the spectrum of the linearization. These eigenvalues, which have the property that  λ= O (1)  as  ɛ→ 0  , govern the stability of the solution on an O (1) time scale. Precise conditions, in terms of the nondimensional parameters, for the stability of symmetric k -spike equilibrium solutions with respect to this class of eigenvalues are obtained. In the low feed-rate regime, it is shown that a large eigenvalue instability leads either to a competition instability, whereby certain spikes in a sequence are annihilated, or to an oscillatory instability (typically synchronous) of the spike amplitudes as a result of a Hopf bifurcation. In the intermediate regime, it is shown that only oscillatory instabilities are possible, and a scaling-law determining the onset of such instabilities is derived. Detailed numerical simulations are performed to confirm the results of the stability theory. It is also shown that there is an equivalence principle between spectral properties of the GS model in the low feed-rate regime and the Gierer–Meinhardt model of morphogenesis. Finally, our results are compared with previous analytical work on the GS model.     

A defect correction method for the time‐dependent Navier‐Stokes equations

A method for solving the time dependent Navier‐Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one‐ and two‐step defect‐correction methods. The spacial error is O(h) for the one‐step defect‐correction method, and O(h²) for the two‐step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection–diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Complete flux scheme for parabolic singularly perturbed differential‐difference equations

In this study, we investigate the concept of the complete flux (CF) obtained as a solution to a local boundary value problem (BVP) for a given parabolic singularly perturbed differential‐difference equation (SPDDE) with modified source term to propose an efficient complete flux‐finite volume method (CF‐FVM) for parabolic SPDDE which is μ‐ and ?‐uniform method where μ, ? are shift and perturbation parameters, respectively. The proposed numerical method is shown to be consistent, stable, and convergent and has been successfully implemented on three test problems.     

Uniform approximation of singularly perturbed reaction‐diffusion problems by the finite element method on a Shishkin mesh

We consider the numerical approximation of singularly perturbed reaction‐diffusion problems over two‐dimensional domains with smooth boundary. Using the h version of the finite element method over appropriately designed piecewise uniform (Shishkin) meshes, we are able to uniformly approximate the solution at a quasi‐optimal rate. The results of numerical computations showing agreement with the analysis are also presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 89–111, 2003     

Linear and non‐linear stability thresholds for thermal convection in a box

We analyse the problem of finding instability thresholds and global non‐linear stability bounds for thermal convection in a linearly viscous fluid in a finite box. The vertical walls are maintained at different temperatures which gives rise to a non‐uniform temperature field in steady state. This problem was previously analysed by Georgescu and Mansutti (Int. J. Non‐Linear Mech. 1999; 34 :603–613). In our work we determine the linear instability threshold to be well above the global stability boundary found by an energy method. Since the perturbed system is not symmetric we expect this to be the case, and our analysis yields a parameter region where possible sub‐critical instabilities may be found. Copyright © 2006 John Wiley & Sons, Ltd.     

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE)

(*)

and Volterra integral equations (VIE)

(**)

by tension spline collocation methods in certain tension spline spaces, where is a small parameter satisfying 0<1, and q₁, q₂, g and K are functions sufficiently smooth on their domains to ensure that Eqs. (*) and (**) posses a unique solution.We give an analysis of the global convergence properties of a new tension spline collocation solution for 0<1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for =1 to the singularly perturbed case.     

Convergence analysis of the LDG method applied to singularly perturbed problems

Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N^‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N²). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A singularly perturbed problem of photon transport in a time‐dependent region

The paper is devoted to the numerical study of a singularly perturbed transport linear integro‐differential equation, in a time‐dependent domain with slab geometry. After a brief summary on existence and uniqueness results for such a model, we test the error between the exact solution and its quasi‐static approximation, which satisfies a simpler equation. Copyright © 2002 John Wiley & Sons, Ltd.     

An hp finite element method for singularly perturbed systems of reactiondiffusion equations

We consider the approximation of a coupled system of two singularly perturbed reaction-diffusion equations by the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We present results on a high order hp finite element scheme which includes elements of size O (εp) and O (μp) near the boundary, where ε, μ are the singular perturbation parameters and p is the degree of the approximating polynomials. Under the assumption of analytic input data, the method yields exponential rates of convergence as p → ∞, independently of ε and μ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cubic spline method for a class of nonlinear singularly-perturbed boundary-value problems

In this paper, we present a numerical method for solving a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. The original second-order problem is reduced to an asymptotically equivalent first-order problem and is solved by a numerical method using a fourth-order cubic spline in the inner region. The method has been analyzed for convergence and is shown to yield anO(h ⁴) approximation to the solution. Some test examples have been solved to demonstrate the efficiency of the method.The authors thank the referee for his helpful comments.     

A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations

A numerical study is made for solving a class of time-dependent singularly perturbed convection–diffusion problems with retarded terms which often arise in computational neuroscience. To approximate the retarded terms, a Taylor’s series expansion has been used and the resulting time-dependent singularly perturbed differential equation is approximated using parameter-uniform numerical methods comprised of a standard implicit finite difference scheme to discretize in the temporal direction on a uniform mesh by means of Rothe’s method and a B-spline collocation method in the spatial direction on a piecewise-uniform mesh of Shishkin type. The method is shown to be accurate of order O(M⁻¹ + N⁻² ln³N), where M and N are the number of mesh points used in the temporal direction and in the spatial direction respectively. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations. Comparisons of the numerical solutions are performed with an upwind and midpoint upwind finite difference scheme on a piecewise-uniform mesh to demonstrate the efficiency of the method.     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.     

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.     

Novel fitted operator finite difference methods for singularly perturbed elliptic convection–diffusion problems in two dimensions

We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ?-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ?-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270].     

Maximum‐norm a posteriori error bounds for a collocation method applied to a singularly perturbed reaction–diffusion problem in three dimensions

Collocation with triquadratic C¹‐splines for a singularly perturbed reaction–diffusion problem in three dimension is studied. A posteriori error bound in the maximum norm is derived for the collocation method on arbitrary tensor‐product meshes which is robust in the perturbation parameter. Numerical results are presented that support our theoretical estimate.     

Scaled discrete derivatives of singularly perturbed elliptic problems

Numerical approximations to the solution of a singularly perturbed elliptic convection–diffusion problem in two space dimensions are generated using a monotone finite difference operator on a tensor product of piecewise‐uniform Shishkin meshes. The bilinear interpolants of these numerical approximations are parameter‐uniformly convergent to the solution of the continuous problem, in the pointwise maximum norm. In this article, discrete approximations to the first derivatives of the solution are shown to be globally first‐order (up to logarithmic factors) uniformly convergent, when the errors are scaled within the analytical layers of the continuous problem. Numerical results are presented to illustrate the theoretical error bounds established in an appropriated weighted C¹–norm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 225–252, 2015     

Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map

We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star‐shaped domain in ?^d, d ≥ 2. Conventional boundary‐based methods require a root search in eigenfrequency k, hence take O(N³) effort per eigenpair found, where N = O(k^d?1) is the number of unknowns required to discretize the boundary. Our method is O(N) faster, achieved by linearizing with respect to k the spectrum of a weighted interior Neumann‐to‐Dirichlet (NtD) operator for the Helmholtz equation. Approximations to the square roots k_j of all O(N) eigenvalues lying in [k ? ?, k], where ? = O(1), are found with O(N³) effort. We prove an error estimate with C independent of k. We present a higher‐order variant with eigenvalue error scaling empirically as O(?⁵) and eigenfunction error as O(?³), the former improving upon the “scaling method” of Vergini and Saraceno. For planar domains (d = 2), with an assumption of absence of spectral concentration, we also prove rigorous error bounds that are close to those numerically observed. For d = 2 we compute robustly the spectrum of the NtD operator via potential theory, Nyström discretization, and the Cayley transform. At high frequencies (400 wavelengths across), with eigenfrequency relative error 10^?10, we show that the method is 10³ times faster than standard ones based upon a root search. © 2014 Wiley Periodicals, Inc.     

Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes

We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.