A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

We consider an arbitrarily sized coupled system of one-dimensional reaction–diffusion problems that are singularly perturbed in nature. We describe an algorithm that uses a discrete Schwarz method on three overlapping subdomains, extending the method in [H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. Comput. Appl. Math. 130 (2001) 231–244] to a coupled system. On each subdomain we use a standard finite difference operator on a uniform mesh. We prove that when appropriate subdomains are used the method produces ε-uniform results. Furthermore we improve upon the analysis of the above-mentioned reference to show that, for small ε, just one iteration is required to achieve the expected accuracy.     

Higher-order compact finite difference method for systems of reaction–diffusion equations

This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction–diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.     

Nonstandard finite difference schemes for reaction‐diffusion equations

We construct finite difference schemes for a particular class of one‐space dimension, nonlinear reaction‐diffusion PDEs. The use of nonstandard finite difference methods and the imposition of a positivity condition constrain the schemes to be explicit and allow the determination of functional relations between the space and time step‐sizes. The general procedure is illustrated by applying it to several important model systems of PDEs © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 201–214, 1999     

Linearized factorization techniques for multidimensional reaction—diffusion equations

An iterative predictor—corrector technique for the elimination of the approximate factorization errors which result from the factorization of linearized θ-methods in multidimensional reaction—diffusion equations is proposed, and its convergence and linear stability are analyzed. Four approximate factorization techniques which do not account for the approximate factorization errors are developed. The first technique uses the full Jacobian matrix of the reaction terms, requires the inversion of, in general, dense matrices, and its approximate factorization errors are second-order accurate in time. The second and third methods approximate the Jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are, however, first-order accurate in time. The fourth approximately factorized method has approximate factorization errors which are second-order accurate in time and requires the inversion of lower and upper triangular matrices. The techniques are applied to a nonlinear, two-species, two-dimensional system of reaction—diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the Jacobian matrix as functions of the allocation of the reaction terms, space and time.     

A reaction–diffusion problem with a reaction front

We state a 1D model with quasi-stationary gas flows approximation for a carbon reactivity test in the production of silicon. The mathematical problem we formulate is a non-linear boundary value problem for a third-order ordinary differential equation with non-linear boundary conditions, which are non-local in time. We prove existence and uniqueness of a classical solution and provide a numerical example. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction–diffusion equations

In this paper, we introduce the concept of norm-to-weak continuous process in a Banach space, and obtain the existence of pullback attractors for this kind of process. Then we give a new method for proving the existence of the pullback attractors. As an application, we obtain the existence of pullback attractors for nonautonomous reaction–diffusion equation in with exponential growth of the external force.     

Numerical exploration of a system of reaction–diffusion equations with internal and transient layers

A reaction pathway for a classical two-species reaction is considered with one reaction that is several orders of magnitudes faster than the other. To sustain the fast reaction, the transport and reaction effects must balance in such a way as to give an internal layer in space. For the steady-state problem, existing singular perturbation analysis rigorously proves the correct scaling of the internal layer. This work reports the results of exploratory numerical simulations that are designed to provide guidance for the analysis to be performed for the transient problem. The full model is comprised of a system of time-dependent reaction–diffusion equations coupled through the non-linear reaction terms with mixed Dirichlet and Neumann boundary conditions. In addition to internal layers in space, the time-dependent problem possesses an initial transient layer in time. To resolve both types of layers as accurately as possible, we design a finite element method with analytic evaluation of all integrals. This avoids all errors associated with the evaluation of the non-linearities and allows us to provide an analytic Jacobian matrix to the implicit time stepping method. The numerical results show that the method resolves the localized sharp gradients accurately and can predict the scaling of the internal layers for the time-dependent problem.     

Asymptotic stability of traveling wave fronts in nonlocal reaction–diffusion equations with delay

This paper is concerned with the asymptotic stability of traveling wave fronts of a class of nonlocal reaction–diffusion equations with delay. Under monostable assumption, we prove that the traveling wave front is exponentially stable by means of the (technical) weighted energy method, when the initial perturbation around the wave is suitable small in a weighted norm. The exponential convergent rate is also obtained. Finally, we apply our results to some population models and obtain some new results, which recover, complement and/or improve a number of existing ones.     

Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating perturbation

We study the normalized difference between the solution u of a reaction–diffusion equation in a bounded interval [0,L], perturbed by a fast oscillating term arising as the solution of a stochastic reaction–diffusion equation with a strong mixing behavior, and the solution of the corresponding averaged equation. We assume the smoothness of the reaction coefficient and we prove that a central limit type theorem holds. Namely, we show that the normalized difference converges weakly in C([0,T];L²(0,L)) to the solution of the linearized equation, where an extra Gaussian term appears. Such a term is explicitly given.     

Global a posteriori error estimates for convection–reaction–diffusion problems

In this note we propose a nonstandard technique for constructing global a posteriori error estimates for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in appropriate weighted energy norms, which measures the overall quality of the approximations, the underlying bilinear form is decomposed into several terms which can be directly computed or easily estimated from above using elementary tools of functional analysis. Several auxiliary parameters are introduced to construct such a splitting and tune the resulting upper error bound. It is demonstrated how these parameters can be chosen in some natural and convenient way for computations so that the weighted energy norm of the error is almost recovered, which shows that the estimates proposed are, in fact, quasi-sharp. The presented methodology is completely independent of numerical techniques used to compute approximate solutions. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g., due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors etc. Moreover, the only constant that appears in the proposed error estimates is of global nature and comes from the Friedrichs–Poincaré inequality.     

Quenching for a reaction–diffusion system with logarithmic singularity

In this paper, we study the quenching phenomenon for a reaction–diffusion system with singular logarithmic source terms and positive Dirichlet boundary conditions. Some sufficient conditions for quenching of the solutions in finite time are obtained, and the blow-up of time-derivatives at the quenching point is verified. Furthermore, under appropriate hypotheses, the non-simultaneous quenching of the system is proved, and the estimates of quenching rate is given.     

Algebraic multigrid preconditioners for the bidomain reaction–diffusion system

The so-called bidomain system is possibly the most complete model for the cardiac bioelectric activity. It consists of a reaction–diffusion system, modeling the intra, extracellular and transmembrane potentials, coupled through a nonlinear reaction term with a stiff system of ordinary differential equations describing the ionic currents through the cellular membrane. In this paper we address the problem of efficiently solving the large linear system arising in the finite element discretization of the bidomain model, when a semiimplicit method in time is employed. We analyze the use of structured algebraic multigrid preconditioners on two major formulations of the model, and report on our numerical experience under different discretization parameters and various discontinuity properties of the conductivity tensors. Our numerical results show that the less exercised formulation provides the best overall performance on a typical simulation of the myocardium excitation process.     

Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities

A class of variable coefficient (1+1)-dimensional nonlinear reaction–diffusion equations of the general form f(x)u_t=(g(x)uⁿu_x)_x+h(x)u^m is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.     

Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.     

Stability and asymptotic behaviour in a reaction–diffusion system

This paper is concerned with some qualitative analysis for a coupled system of five reaction–diffusion equations which arises from a physiology model. The uniform boundedness of the time-dependent solution is obtained under various boundary conditions. Sufficient conditions are also given to ensure the asymptotic stability of the non-negative steady-state solutions under Dirichlet or Robin boundary condition for each component. Under homogeneous Neumann boundary condition for some components the time-dependent solution is proven to converge to a constant steady state determined by the initial functions.     

Pointwise a priori bounds for a strongly coupled system of reaction–diffusion equations with a balance law

We consider a reaction–diffusion system with a full matrix of diffusion coefficients satisfying a balance law on a bounded domain with no-flux boundary conditions. We demonstrate that global solutions exist for polynomial reaction terms provided some conditions on the diffusion coefficients are satisfied. The proof makes use of comparison results and Solonnikov's estimates concerning linear parabolic equations in Banach spaces. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Multilevel a posteriori error analysis for reaction–convection–diffusion problems

We present a new approach to the a posteriori error analysis of stable Galerkin approximations of reaction–convection–diffusion problems. It relies upon a non-standard variational formulation of the exact problem, based on the anisotropic wavelet decomposition of the equation residual into convection-dominated scales and diffusion-dominated scales. The associated norm, which is stronger than the standard energy norm, provides a robust (i.e., uniform in the convection limit) control over the streamline derivative of the solution. We propose an upper estimator and a lower estimator of the error, in this norm, between the exact solution and any finite dimensional approximation of it. We investigate the behaviour of such estimators, both theoretically and through numerical experiments. As an output of our analysis, we find that the lower estimator is quantitatively accurate and robust.     

On the convergence of basic iterative methods for convection–diffusion equations

In this paper we analyze convergence of basic iterative Jacobi and Gauss–Seidel type methods for solving linear systems which result from finite element or finite volume discretization of convection–diffusion equations on unstructured meshes. In general the resulting stiffness matrices are neither M‐matrices nor satisfy a diagonal dominance criterion. We introduce two newmatrix classes and analyse the convergence of the Jacobi and Gauss–Seidel methods for matrices from these classes. A new convergence result for the Jacobi method is proved and negative results for the Gauss–Seidel method are obtained. For a few well‐known discretization methods it is shown that the resulting stiffness matrices fall into the new matrix classes. Copyright © 1999 John Wiley & Sons, Ltd.     

A two‐grid method for mixed finite‐element solution of reaction‐diffusion equations

We present a scheme for solving two‐dimensional, nonlinear reaction‐diffusion equations, using a mixed finite‐element method. To linearize the mixed‐method equations, we use a two grid scheme that relegates all the Newton‐like iterations to a grid Δ_H much coarser than the original one Δ_h, with no loss in order of accuracy so long as the mesh sizes obey . The use of a multigrid‐based solver for the indefinite linear systems that arise at each coarse‐grid iteration, as well as for the similar system that arises on the fine grid, allows for even greater efficiency. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 317–332, 1999     

Global boundedness of solutions to a reaction–diffusion system

We obtain in this paper the global boundedness of solutions to a Fujita‐type reaction–diffusion system. This global boundedness results from diffusion effect, homogeneous Dirichlet boundary value conditions and appropriate reactions. Copyright © 1999 John Wiley & Sons, Ltd.