Shadow systems and attractors in reaction-diffusion equations

For a pair of reaction diffusion equations with one diffusion coefficient very large, there is associated a reaction diffusion equation coupled with an ordinary differential equation (the shadow system) with nonlocal effects which has the property that it contains all of the essential dynamics of the original equations. Key words: Attractors, shadow systems, reaction-diffusion equations     

The blow-up for weakly coupled reaction-diffusion systems

In this paper we consider a weakly coupled parabolic system with nonnegative exponents in the forcing functions. We find the conditions which result in blow-up in finite time. Also, we obtain the blow-up rate.

     

Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria

We consider the asymptotic behavior for large time of solutions to reaction-diffusion systems modeling reversible chemical reactions. We focus on the case where multiple equilibria exist. In this case, due to the existence of so-called "boundary equilibria", the analysis of the asymptotic behavior is not obvious. The solution is understood in a weak sense as a limit of adequate approximate solutions. We prove that this solution converges in L^1 toward an equilibrium as time goes to infinity and that the convergence is exponential if the limit is strictly positive.     

Spatial Patterns for reaction-diffusion systems with conditions described by inclusions

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.     

Resilience in reaction-diffusion systems

Reaction-diffusion systems with zero-flux Neumann boundariesare widely used to model various kinds of interaction in, forexample, the scientific fields of ecology, biology, chemistry,medicine and industry. The physical systems within these fieldsare often known to be (conditionally or unconditionally) resilientwith respect to shocks, disturbances or catastrophies in theimmediate environment. In order to be good mathematical modelsof such situations the reaction-diffusion systems must havethe same resilient or asymptotic behaviour as that of the physicalsituation. Three fundamentally different kinds of reaction termsare usually distinguished according to the entry signs of thereaction Jacobian: mutualism, mixed (predator-prey) interactionand competition. The asymptotic stability (in the Poincarésense) of mutualistic systems has already been studied extensively,but the results cannot be generalized (globally) to the othertwo fundamental types, which are not order-preserving. A partial(local) generalization is, however given here for these twotypes, involving simple Jacobian inequalities and knowledge(often prompted by the underlying physical situation) of invariantsets in solution space. The return time of resilient systemsand the approach rate of asymptotically stable solutions arealso estimated.     

SBA splitting for approximation of invariants in time-dependent Hamiltonian systems

Most physical phenomena are described by time-dependent Hamiltonian systems with qualitative features that should be preserved by numerical integrators used for approximating their dynamics. The initial energy of the system together with the energy added or subtracted by the outside forces, represent a conserved quantity of the motion. For a class of time-dependent Hamiltonian systems [8] this invariant can be defined by means of an auxiliary function whose dynamics has to be integrated simultaneously with the system’s equations. We propose splitting procedures featured by a SB³A property that allows to construct composition methods with a reduced number of determining order equations and to provide the same high accuracy for both the dynamics and the preservation of the invariant quantity.     

A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction-diffusion problems

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction-diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results.     

A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media

A Galerkin approach for a class of multiscale reaction-diffusion systems with nonlinear coupling between the microscopic and macroscopic variables is presented. This type of models are obtained e.g. by upscaling of processes in chemical engineering (particularly in catalysis), biochemistry, or geochemistry. Exploiting the special structure of the models, the functions spaces used for the approximation of the solution are chosen as tensor products of spaces on the macroscopic domain and on the standard cell associated to the microstructure. Uniform estimates for the finite dimensional approximations are proven. Based on these estimates, the convergence of the approximating sequence is shown. This approach can be used as a basis for the numerical computation of the solution.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

TARGET PATTERNS AND SPIRAL WAVES IN REACTION-DIFFUSION SYSTEMS WITH LIMIT CYCLE KINETICSAND WEAK DIFFUSION

1.IntroductionTargetpatternsandspiralwavesarecommonlyobservedincertainmodelsofchemicalandbiologicalsystemssuchastheBelousov-ZhabotinskiireactionandthesocialamoebasDictyosteliumdiscoideium(of.11--4]andthereferencestherein).Thesesystemsaregovernedbyachemicalorbiologicalreactionandspatialdiffusion,i.e.reaction-diffusionequations.Generallyspeaking,atargetpatternisasetofconcentricringsofconstantconcentrationeachmovingoutward.Alonganyradialline,thepatternbehavesasymptoticallylikeaperiodictravellin…     

An almost third order finite difference scheme for singularly perturbed reaction-diffusion systems

This paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction-diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given.     

Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem

Consider the problem with homogeneous Neumann boundary condition in a bounded smooth domain in . The whole range is treated. The Galerkin finite element method is used on a globally quasi-uniform mesh of size ; the mesh is fixed and independent of .

A precise analysis of how the error at each point depends on and is presented. As an application, first order error estimates in , which are uniform with respect to , are given.

     

Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction

Summary. In this paper, we perform the numerical analysis of operator splitting techniques for nonlinear reaction-diffusion systems with an entropic structure in the presence of fast scales in the reaction term. We consider both linear diagonal and quasi-linear non-diagonal diffusion; the entropic structure implies the well-posedness and stability of the system as well as a Tikhonov normal form for the nonlinear reaction term [23]. It allows to perform a singular perturbation analysis and to obtain a reduced and well-posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution. We then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step t thus leading to the failure of the usual splitting analysis techniques. We define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales. All the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level. We finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis.Mathematics Subject Classification (2000): 65M12, 35K57, 35B25, 35Q80, 34E15, 80A32, 92E20     

LDG method for reaction-diffusion dynamical systems with time delay

In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction-diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.     

Convergences of splitting iterative methods for symmetric indefinite linear systems

In this paper, we generalize the saddle point problem to general symmetric indefinite systems, we also present a kind of convergent splitting iterative methods for the symmetric indefinite systems. A special divergent splitting is introduced. The sufficient condition is discussed that the eigenvalues of the iteration matrix are real. The spectral radius of the iteration matrix is discussed in detail, the convergence theories of the splitting iterative methods for the symmetric indefinite systems are obtained. Finally, we present a preconditioner and discuss the eigenvalues of preconditioned matrix.     

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionu _t=εDu _xx+(A+εA ₁)u+F(u),u ∈ ℝ². These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0. One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε ^−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.     

A higher order method for input-affine uncertain systems

Uncertainty is unavoidable in modeling dynamical systems and it may be represented mathematically by differential inclusions. In the past, we proposed an algorithm to compute validated solutions of differential inclusions; here we provide several theoretical improvements to the algorithm, including its extension to piecewise constant and sinusoidal approximations of uncertain inputs, updates on the affine approximation bounds and a generalized formula for the analytical error. The approach proposed is able to achieve higher order convergence with respect to the current state-of-the-art. We implemented the methodology in Ariadne, a library for the verification of continuous and hybrid systems. For evaluation purposes, we introduce ten systems from the literature, with varying degrees of nonlinearity, number of variables and uncertain inputs. The results are hereby compared with two state-of-the-art approaches to time-varying uncertainties in nonlinear systems.     

A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + h k+1 + H 2k+2 d/2 ) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.     

Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem

A non-monotone FEM discretization of a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers is considered. The method is shown to be maximum-norm stable although it is not inverse monotone. Both a priori and a posteriori error bounds in the maximum norm are derived. The a priori result can be used to deduce uniform convergence of various layer-adapted meshes proposed in the literature. Numerical experiments complement the theoretical results. AMS subject classification (2000)  65L10, 65L50, 65L60