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.     

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

A semilinear reaction–diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter e²{\varepsilon^2} , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult. To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub- and super-solutions. Convergence results are deduced that depend on the relative sizes of e{\varepsilon} and N, where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton’s method to compute a discrete solution are discussed.     

On the convergence of the Dirichlet grid problem with a singularity for a singularly perturbed convection–diffusion equation

The Dirichlet problem for a singulary perturbed convection–diffusion equation in a rectangle when a discontinuity at the flow exit the first derivative of the boundary condition gives rise to an inner layer for the solution. On piecewise-uniform Shishkin grids that condense near regular and characteristic layers, the solution obtained using the classical five-point difference scheme with a directed difference is shown to converge with respect to the small parameter to solve the original problem in the grid norm L _∞ ^h almost with the first order. This theoretical result is confirmed via numerical analysis.     

Computer difference scheme for a singularly perturbed elliptic convection–diffusion equation in the presence of perturbations

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε, ε ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge -uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if N ₁ ^-1 N ₂ ^-1 ? ε, where N ₁ and N ₂ are the numbers of grid intervals in x and y, respectively, the scheme is not -uniformly well-conditioned or ε-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on ε, N ₁,N ₂, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as N ₁,N ₂ → ∞, ε ∈ (0,1]. The difference schemes constructed in the presence of the indicated perturbations that converges as N ₁,N ₂ → ∞ for fixed ε, ε ∈ (0,1, is called a computer difference scheme. Schemes converging ε-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.     

A second order scheme for a time-dependent,singularly perturbed convection–diffusion equation

We consider a numerical scheme for a one-dimensional, time-dependent, singularly perturbed convection–diffusion problem. The problem is discretized in space by a standard finite element method on a Bakhvalov–Shishkin type mesh. The space error is measured in an L² norm. For the time integration, the implicit midpoint rule is used. The fully discrete scheme is shown to be convergent of order 2 in space and time, uniformly in the singular perturbation parameter.     

Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems

A procedure for the construction of robust, upper bounds for the error in the finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method.     

Numerical computation of sharp travelling waves of a degenerate diffusion–reaction equation arising in biofilm modelling

Many degenerate diffusion–reaction equations permit sharp travelling wave solutions that describe the propagation of an interface with finite speed. If the equation is at least double degenerate, the derivative of the travelling wave solution can blow up at the interface, which poses considerable challenges for the computation of the travelling wave speed. We propose a numerical method for this problem that is based on the idea to approximate the multiple degenerate problem by a family of simple degenerate problems. For the latter we propose an interval-bracketing algorithm based on the theory of Sanchez-Garduno and Maini. The travelling wave speed of the original problem is obtained as the limit of the travelling wave speeds of the auxiliary problems. The performance of the method is investigated in a numerical simulation experiment for a problem that arises in the mathematical modelling of biofilm processes.     

Convergence of a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection–diffusion equation in 2D

A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.     

Numerical analysis of a strongly coupled system of two singularly perturbed convection–diffusion problems

A system of two coupled singularly perturbed convection–diffusion ordinary differential equations is examined. The diffusion term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The problem does not satisfy a conventional maximum principle. Its solution is decomposed into regular and layer components. Bounds on the derivatives of these components are established that show explicitly their dependence on the small parameter. A numerical method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh is shown to generate numerical approximations that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum norm.      

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.     

Standard finite difference scheme for a singularly perturbed elliptic convection–diffusion equation on a rectangle under computer perturbations

A singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε (ε ∈ (0, 1]) is considered on a rectangle. As applied to this equation, a standard finite difference scheme on a uniform grid is studied under computer perturbations. This scheme is not ε-uniformly stable with respect to perturbations. The conditions imposed on a “computing system” are established under which a converging standard scheme (referred to as a computer difference scheme) remains stable.     

Speed of wave propagation for a nonlocal reaction–diffusion equation

ABSTRACT

A nonlocal reaction–diffusion equation arising in various applications is studied. The speed of traveling waves is determined by means of a minimax representation. It is used to obtain the wave speed estimates and asymptotic values.     

A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation

We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection–reaction–diffusion equation. Equations of this type arise in many contexts, such as for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet–Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically.     

Spreading speeds of a partially degenerate reaction–diffusion system in a periodic habitat

This paper is devoted to the study of the spreading speeds of a partially degenerate reaction–diffusion system with monostable nonlinearity in a periodic habitat. We first obtain sufficient conditions for the existence of principal eigenvalues in the case where solution maps of the associated linear systems lack compactness, and prove a threshold type result on the global dynamics for the periodic initial value problem. Then we establish the existence and computational formulae of spreading speeds for the general initial value problem. It turns out that the spreading speed is linearly determinate.     

On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C²$C^2$ classical regularity, but also the existence of discontinuous entropy traveling wave solutions.