Perturbed Reaction Diffusion Research Articles

Pulse dynamics in a three-component system: Stability and bifurcations

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction–diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction–diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction–diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.

Soft and Hard Wall in a Stochastic Reaction Diffusion Equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a “soft” repulsion from the boundary. We finally show how a “hard” repulsion can be obtained by an extra diffusive scaling.

A class of nonlinear nonlocal singularly perturbed reaction diffusion systems

The singularly perturbed initial boundary value problems for the reaction diffusion system are raised. Firstly, under suitable conditions, using a iteration technique, the differential inequalities theorem is constructed and introducing two auxiliary functions the existence and uniqueness theorem of solution for the basic reaction diffusion system is proved. Using the singularly perturbed method the formal asymptotic expressions of the solution are constructed with power series theory. By using the comparison theorem the existence and its asymptotic behavior of solution for the original problem are studied. Finally, using method of estimate inequalities, the structure of solutions for the problem is discussed thoroughly in three cases and asymptotic solution of the original problem is given. The asymptotic behavior of solution in the three cases is proved.

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

AbstractThe numerical approximation by a lower‐order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving semisingular perturbation problems. The quasi‐optimal‐order error estimates are proved in the ε‐weighted H1‐norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε‐weighted H1‐norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.

Finite elements scheme in enriched subspaces for singularly perturbed reaction–diffusion problems on a square domain

In this article, we discuss reaction-diffusion problems which produce ordinary boundary layers and elliptic corner layers. Using the classical polynomial Q1-finite elements spaces enriched with the so-called boundary layer elements which absorb the singularities due to the boundary and corner layers we are able to attain high numerical accuracies. We essentially obtain � -uniform approximation errors in a weighted energy norm with significant simplifications in the numerical implemen- tations; here we do not use mesh refinements.

Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

The bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems. In each case, the quasi-optimal order error estimates are proved in the ε -weighted H 1 -norm uniformly in singular perturbation parameter ε , up to a logarithmic factor. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε -weighted H 1 -norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.

CUBIC SPLINE ADAPTIVE WAVELET SCHEME TO SOLVE SINGULARLY PERTURBED REACTION DIFFUSION PROBLEMS

In this paper, the collocation method proposed by Cai and Wang1 has been reviewed in detail to solve singularly perturbed reaction diffusion equation of elliptic and parabolic types. The method is based on an interpolating wavelet transform using cubic spline on dyadic points. Adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples are presented for elliptic and parabolic problems. The purposed method comes up as a powerful tool for studying singular perturbation problems in term of effective grid generation and CPU time.

Algebraically decaying pulses in a Ginzburg–Landau system with a neutrally stable mode

In this paper, we study the existence and stability of pulse solutions in a system with interacting instability mechanisms, which is described by a Ginzburg–Landau equation for an A-mode, coupled to a diffusion equation for a B-mode. Our main question is whether this coupling may stabilize solutions of the Ginzburg–Landau equation that are unstable when the interactions with the neutrally stable B-mode are not included in the model. The spatially homogeneous B-mode is supposed to be neutrally stable. This implies that the pulse solutions cannot decay exponentially, but must decay with an algebraic rate as x → ±∞. As a consequence, the methods that exist in the literature by which the stability of pulses in singularly perturbed reaction–diffusion systems can be studied need to be extended. This results in an ‘algebraic NLEP approach’, which is expected to be relevant beyond the setting of this paper. As in the case of a (weakly) stable B-mode (Doelman et al 2004 J. Nonlin. Sci. 14 237–78) we establish by the application of this approach that the B-mode indeed introduces a mechanism that may stabilize pulses that are unstable when the interactions with the B-mode are not taken into account.

A posteriori H1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H 1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Three modifications of these estimators are introduced and discussed. Much attention is given to the performance of the error estimator in numerical experiments. This helps to identify those estimators that are suitable for practical applications.

A multiscale a posteriori error estimate

We introduce a hierarchic a posteriori error estimate for singularly perturbed reaction–diffusion problems. The estimator is based on a Petrov–Galerkin method in which the trial space is enriched with nonpolynomial functions or multiscale functions. We study the equivalence between the a posteriori estimate and the exact error in the energy norm. Moreover, we prove a relationship between the hierarchic estimator and an explicit residual estimator. The approach provides accurate estimates for the boundary layer regions which is confirmed by numerical experiments.

A singularly perturbed reaction–diffusion problem with inhomogeneous environment

Reaction–diffusion problems have been used to describe pattern formation in developmental biology and material science. I study the existence and stability of a singularly perturbed reaction–diffusion problem with inhomogeneous environment in one-dimensional space domain and also high-dimensional space domain.

A uniformly accurate spline collocation method for a normalized flux

We are concerned with a two-point boundary value problem for a semilinear singularly perturbed reaction–diffusion equation with a singular perturbation parameter ε. Our goal is to construct global ε-uniform approximations of the solution y( x) and the normalized flux P( x)= ε(d/d x) y( x), using the collocation with the classical quadratic splines u( x)∈ C 1( I) on a slightly modified piecewise uniform mesh of Shishkin type. The constructed approximate solution and normalized flux converge ε-uniformly with the rate O(n −2 ln 2 n) and O(n −1 ln n) , respectively, on the Shishkin-type mesh, and with O(n −1 ln −2 n) and O( ln −3 n) when the mesh has to be modified. We present numerical experiments in support of these results.

On monotone iterative methods for a nonlinear singularly perturbed reaction–diffusion problem

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed reaction–diffusion problem. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process. The rate of convergence of the monotone Schwarz method is estimated. Numerical experiments are presented.

A finite element analysis of a coupled system of singularly perturbed reaction–diffusion equations

We consider a system of two coupled reaction–diffusion equations. When the parameters multiplying the second-order derivatives in the equations are small, their solutions exhibit boundary layers. Moreover, when the parameters are of different magnitudes, two distinct but overlapping boundary layers are present. We study a finite element discretization on general layer-adapted meshes including the frequently studied Shishkin mesh and the Bakhvalov mesh. Supporting numerical results are presented.

A note on the energy norm for a singularly perturbed model problem

This paper considers a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research.

A numerical method for a system of singularly perturbed reaction–diffusion equations

A Dirichlet problem for a system of two coupled singularly perturbed reaction–diffusion ordinary differential equations is examined. A numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. Numerical results are presented, which illustrate the theoretical results.

Uniformly convergent high order finite element solutions of a singularly perturbed reaction–diffusion equation using mesh equidistribution

We study the numerical approximation of a singularly perturbed reaction–diffusion equation using a pth order Galerkin finite element method on a non-uniform grid. The grid is constructed by equidistributing a strictly positive monitor function which is a linear combination of a constant floor and a power of the second derivative of a representation of the boundary layers—obtained using a suitable decomposition of the analytical solution. By the appropriate selection of the monitor function parameters we prove that the numerical solution is insensitive to the size of the singular perturbation parameter and achieves the optimal rate of convergence with respect to the mesh density.

On a uniformly accurate finite difference approximation of a singularly perturbed reaction–diffusion problem using grid equidistribution

We examine the convergence properties of a finite difference approximation of a singularly perturbed reaction–diffusion boundary value problem using a nonuniform grid. The grid is based on the equidistribution of a positive monitor function that is a linear combination of a constant floor and a power of the second derivative of the solution. Analysis shows how the monitor function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter. The use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids. Numerical results are given that confirm the uniform convergence rates.

Convergence and superconvergence analysis of finite element methods on highly nonuniform anisotropic meshes for singularly perturbed reaction–diffusion problems

The standard finite element methods on one type of highly nonuniform rectangular meshes, which is different from the extensively discussed Shishkin type meshes, are considered for solving the singular perturbation problem − div(a∇u)+bu=f , where the diagonal tensor a=(ε 2,1) or a=(ε 2,ε 2) . Global uniform convergence rates of O (N −2) for u in the L 2 -norm are obtained in both cases for bilinear rectangular finite elements, where N is the number of intervals in both the x- and y-directions. The pointwise interior (away from the boundary layers) convergence rates of O (N −1) for u are also proved. Global superconvergence rates of O (N −2) in the L 2 -norm for a 1/2∇u are obtained by a postprocessing. Numerical experiments supporting the theoretical analysis show that the new type anisotropic meshes give equivalent performance as the Shishkin type meshes.

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.