Perturbed Reaction-diffusion Equations Research Articles

Difference Scheme on a Non-Uniform Mesh for Singularly Perturbed Reaction Diffusion Equations with Integral Boundary Condition

In this paper we consider singularly perturbed reaction diffusion equations with integral boundary condition. A numerical method based on finite difference scheme on Shishkin mesh is presented. This method is proved to be almost second order convergent. An error estimate is derived in the discrete norm. Numerical results are presented, which validate the theoretical results.

Convergence of a finite element method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

A finite element method is applied on a Bakhvalov-type mesh to solve a singularly perturbed reaction–diffusion problem whose solution exhibits boundary layers. A uniform convergence order of O(N−(k+1)+ε1/2N−k) is proved, where k is the order of piecewise polynomials in the finite element method, ε is the diffusion parameter and N is the number of partitions in each direction. Numerical experiments support this theoretical result.

Two-grid algorithm for a system of singularly perturbed reaction-diffusion equations on Shishkin mesh

A system of two coupled singularly perturbed reaction-diffusion equations is considered. A cascadic two-grid algorithm of high accuracy based on a uniform with respect to small parameters convergent difference scheme for such problem has been developed. To increase the accuracy of the difference solution by an order, the Richardson extrapolation method was applied. The results of some numerical experiments are discussed.

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

<p style='text-indent:20px;'>In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.</p>

A Uniformly Convergent Numerical Scheme for a Coupled System of Singularly Perturbed Reaction-Diffusion Equations

Here, we study the numerical solution of singularly perturbed system of two-point boundary-value problems of reaction-diffusion type. The solutions of these problems exhibit twin overlapping exponential boundary layers. To obtain the numerical solutions of these problems, we apply the nonsymmetric discontinuous Galerkin FEM with interior penalties (NIPG method) on layer-adapted piecewise-uniform Shishkin mesh. Also, we have proved that the proposed method is uniformly convergent with order in the weighted DG energy norm, where is the degree of the piecewise polynomial in the finite element space. Numerical results are presented to support the theoretical results.

A robust second order numerical method for a weakly coupled system of singularly perturbed reaction-diffusion problem with discontinuous source term

In this paper, a fitted mesh numerical method on Shishkin mesh is proposed to solve a weakly coupled system of two singularly perturbed reaction-diffusion equations containing equal diffusion parameters with discontinuous source terms. This method uses the standard centred finite difference scheme constructed on piecewise-uniform Shishkin mesh with the average of the source terms on either side of the point of discontinuity and then the problem is solved by an iterative procedure. An error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to confirm the theoretical results and compares well with the existing results.

Asymptotic analysis for elliptic equations with Robin boundary condition

We investigate the boundary layers of a singularly perturbed reaction-diffusion equation in a 3D channel domain. The equation is supplemented with a Robin boundary condition especially when the smooth function on the boundary, appearing in the Robin boundary condition, depends on the perturbation parameter. By constructing an explicit function, called corrector, which describes behavior of the perturbed solution near the boundary, we obtain an asymptotic expansion of the perturbed solution as the sum of the corresponding limit solution and the corrector, and show the convergence in L 2 of the perturbed solution to the limit solution as the perturbation parameter tends to zero.

A robust second order numerical method for a weakly coupled system of singularly perturbed reaction-diffusion problem with discontinuous source term

In this paper, a fitted mesh numerical method on Shishkin mesh is proposed to solve a weakly coupled system of two singularly perturbed reaction-diffusion equations containing equal diffusion parameters with discontinuous source terms. This method uses the standard centred finite difference scheme constructed on piecewise-uniform Shishkin mesh with the average of the source terms on either side of the point of discontinuity and then the problem is solved by an iterative procedure. An error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to confirm the theoretical results and compares well with the existing results.

Multilevel Schwarz preconditioners for singularly perturbed symmetric reaction-diffusion systems

We present robust and highly parallel multilevel non-overlapping Schwarz preconditioners, to solve an interior penalty discontinuous Galerkin finite element discretization of a system of steady state, singularly perturbed reaction-diffusion equations with a singular reaction operator, using a GMRES solver. We provide proofs of convergence for the two-level setting and the multigrid V-cycle as well as numerical results for a wide range of regimes.

How accurate are finite elements on anisotropic triangulations in the maximum norm?

In Kopteva (2014) a counterexample of an anisotropic triangulation was given on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. This example was given in the context of a singularly perturbed reaction–diffusion equation. In this paper, we present further examples of unanticipated pointwise convergence behaviour of Lagrange finite elements on anisotropic triangulations. In particular, we show that linear finite elements may exhibit lower than expected orders of convergence for the Laplace equation, as well as for certain singular equations, and their accuracy depends not only on the linear interpolation error, but also on the mesh topology. Furthermore, we demonstrate that pointwise convergence rates which are worse than one might expect are also observed when higher-order finite elements are employed on anisotropic meshes. A theoretical justification will be given for some of the observed numerical phenomena.

Uniformly convergent novel finite difference methods for singularly perturbed reaction–diffusion equations

AbstractIn this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction.

Numerical Method for a System of Singularly Perturbed Reaction Diffusion Equations with Integral Boundary Conditions

A class of system of singularly perturbed reaction diffusion equations with integral boundary conditions is considered. A numerical method based on a finite difference scheme on a Shishkin mesh is presented. The method suggested is of almost second order convergent. An error estimate is derived in the discrete norm. Numerical examples presented validate the theoretical estimates.

Numerical solutions of a system of singularly perturbed reaction-diffusion problems

This paper addresses the numerical approximation of solutions to a coupled system of singularly perturbed reaction-diffusion equations. The components of the solution exhibit overlapping boundary and interior layers. Sinc procedure can control the oscillations in computed solutions at boundary layer regions naturally because the distribution of Sinc points is denser at near the boundaries. Also the obtained results show that the proposed method is applicable even for small perturbation parameter as ? = 2-30. The convergence analysis of proposed technique is discussed, it is shown that the approximate solutions converge to the exact solutions at an exponential rate. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the method.

Asymptotic Approximation of the Solution of the Reaction-Diffusion-Advection Equation with a Nonlinear Advective Term

We consider a solution in a moving front form of the initial-boundary value problem for a singularly perturbed reaction-diffusion equation in a band with periodic conditions in one of the variables. Interest in solutions of the front type is associated with combustion problems or nonlinear acoustic waves. In the domain of the function which describes the moving front there is a subdomain where the function has a large gradient. This subdomain is called the internal transition layer. Boundary value problems with internal transition layers have a natural small parameter that is equal to the ratio of the transition layer width to the width of the region under consideration. The presence of a small parameter at the highest spatial derivative makes the problem singularly perturbed. The numerical solution of such problems meets certain difficulties connected with the choice of grids and initial conditions. To solve these problems the use of analytical methods is especially successful. Asymptotic analysis which uses Vasilieva’s algorithm was carried out in the paper. That made it possible to obtain an asymptotic approximation of the solution, which can be used as an initial condition for a numerical algorithm. We also determined the conditions for the existence of a front type solution. In addition, the analytical methods used in the paper make it possible to obtain in an explicit form the front motion equation approximation. This information can be used to develop mathematical models or numerical algorithms for solving boundary value problems for the reaction-diffusion-advection type equations.

Numerical Simulation of Singularly Perturbed Reaction-Diffusion Equation Using Finite Element Method

This article deals with the study of sign-changing solutions of the nonlinear singularly perturbed reaction-diffusion equation. Sign changing solutions of the nonlinear problem do not appear to have been previously studied in detail computationally, and it is hoped that this paper will help to provide a new idea in this direction. A variant of Newton’s method having tenth order of convergence has been established to linearize the nonlinear system of equations. Examples of the nonlinear problem having nonlinearities in homogeneous/nonhomogeneous form are considered to show the existence of solutions.

Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations

In this paper, we study the relations between the long-time dynamical behavior of the perturbed reaction-diffusion equations and the exact reaction-diffusion equations with concave and convex nonlinear terms and prove that bounded sets of solutions of the perturbed reaction-diffusion equations converges to the trajectory attractor $ \mathscr{U}_0 $ of the exact reaction-diffusion equations when $ t\rightarrow\infty $ and $ \varepsilon\rightarrow 0^+. $ In particular, we show that the trajectory attractor $ \mathscr{U}_ \varepsilon $ of the perturbed reaction-diffusion equations converges to the trajectory attractor $ \mathscr{U}_0 $ of the exact reaction-diffusion equations when $ \varepsilon\rightarrow 0^+. $ Moreover, we derive the upper and lower bounds of the fractal dimension for the global attractor of the perturbed reaction-diffusion equations.

Robust coupling of DPG and BEM for a singularly perturbed transmission problem

We consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the Laplacian in the exterior, connected through standard transmission conditions. We establish a DPG scheme coupled with Galerkin boundary elements for its discretization, and prove its robustness for the field variables in so-called balanced norms. Our coupling scheme is the one from Führer et al., (2016), adapted to the singularly perturbed case by using the scheme from Heuer and Karkulik (2015). Essential feature of our method is that optimal test functions have to be computed only locally. We report on various numerical experiments in two dimensions.

A robust numerical method for a control problem involving singularly perturbed equations

We consider an unconstrained linear–quadratic optimal control problem governed by a singularly perturbed convection–reaction–diffusion equation. We discretize the optimality system by using standard piecewise bilinear finite elements on the graded meshes introduced by Durán and Lombardi in (Duŕan and Lombardi 2005, 2006). We prove convergence of this scheme. In addition, when the state equation is a singularly perturbed reaction–diffusion equation, we derive quasi-optimal a priori error estimates for the approximation error of the optimal variables on anisotropic meshes. We present several numerical experiments when the state equation is both a reaction–diffusion and a convection–reaction–diffusion equation. These numerical experiments reveal a competitive performance of the proposed solution technique.

Difference scheme of highest accuracy order for a singularly perturbed reaction–diffusion equation based on the solution decomposition method

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction–diffusion equation. For this problem, a new approach is developed in order to construct difference schemes whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter e, e ∈ (0, 1] (i.e., e-uniformly) with order of accuracy significantly greater than the ultimate achievable accuracy order for the Richardson method on piecewise uniform grids. The main point of this approach is that uniform grids are used to solve grid subproblems for the regular and singular components of the discrete solution. Using the asymptotic construction technique, a basic difference scheme of the solution decomposition method is constructed that converges e-uniformly in the maximum norm at the rate O(N −2 ln2 N), where N + 1 is the number of nodes in the uniform grids used. The Richardson extrapolation technique on three embedded grids is applied to the basic scheme of the solution decomposition method. As a result, we have constructed the Richardson scheme of the solution decomposition method with highest accuracy order. The solution of this scheme converges e-uniformly in the maximum norm at the rate O(N −6 ln6 N).

A discontinuous Galerkin least-squares finite element method for solving coupled singularly perturbed reaction–diffusion equations

A discontinuous Galerkin least-squares finite element method is proposed to solve coupled reaction–diffusion equations with singular perturbations. This method produces solutions without numerical oscillations when uniform meshes are used. Numerical examples are provided to demonstrate the efficiency of the method.