A parameter robust numerical method for a two dimensional reaction-diffusion problem.

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques

Boundary layers in a two-point boundary value problem with a caputo fractional derivative

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Caputo fractional derivative of order ¿ with 1 < ¿ < 2. Writing ¿ for the solution of the problem, it is known that typically ¿¿¿(¿) blows up as ¿ ¿ 0. A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: ¿ may exhibit a boundary layer at ¿ = 1 when ¿ is near 1. The conditions on the data of the problem under which this layer appears are investigated by first solving the constant-coefficient case using Laplace transforms, determining precisely when a layer is present in this special case, then using this information to enlighten our examination of the general variable-coefficient case (in particular, in the construction of a barrier function for ¿). This analysis proves that usually no boundary layer can occur in the solution ¿ at ¿ = 0, and that the quantity ¿ = max¿¿[0,1] ¿(¿), where ¿ is the coefficient of the first-order term in the differential operator, is critical: when¿ < 1,noboundarylayerispresentwhen¿isnear1,butwhen¿ = 1thenaboundarylayerat¿ = 1 is possible. Numerical results illustrate the sharpness of most of our results

Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

High order schemes for reaction-diffusion singularly perturbed systems

In this paper we are interested in solving e¿ciently a singularly per-turbed linear system of di¿erential equations of reaction-di¿usion type. Firstly, anon–monotone ¿nite di¿erence scheme of HODIE type is constructed on a Shishkinmesh. The previous method is modi¿ed at the transition points such that an inversemonotone scheme is obtained. We prove that if the di¿usion parameters are equal itis a third order uniformly convergent method. If the di¿usion parameters are di¿er-ent some numerical evidence is presented to suggest that an uniformly convergentscheme of order greater than two is obtained. Nevertheless, the uniform errors arebigger and the orders of uniform convergence are less than in the case correspondingto equal di¿usion parameters

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

AbstractThis 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

Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation

A reaction-diffusion problem with a Caputo time derivative of order = 2 (0; 1) is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time t = 0, and sharp point wise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis

An efficient numerical scheme for 1D parabolic singularly perturbed problems with an interior and boundary layers

In this paper we consider a 1D parabolic singularly perturbed reaction-convection-diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter e2) and the convection term (multiplied by the parameter µ) in the differential equation (e¿(0, 1], µ¿0, 1], µ=e). Moreover, the convective term degenerates inside the spatial domain, and also the source term has a discontinuity of first kind on the degeneration line. In general, for sufficiently small values of the diffusion and the convection parameters, the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain. We study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme, which combines the implicit Euler method, defined on a uniform mesh, to discretize in time, together with the classical upwind finite difference scheme, defined on an appropriate nonuniform mesh of Shishkin type, to discretize in space. The numerical scheme converges in the maximum norm uniformly in e and µ, having first order in time and almost first order in space. Illustrative numerical results corroborating in practice the theoretical results are showed

Development of POD-based Reduced Order Models applied to shallow water equations using augmented Riemann solvers

Reduced-order models (ROMs) based on the proper orthogonal decomposition have been proposed to reduce the computational resources required by the full-order models (FOMs) to approximate partial differential equations. In this paper a Roe-based ROM is developed to solve the shallow water equations in presence of source terms more efficiently than the Roe-based FOM. The well-balanced property and other numerical corrections such as the entropy fix and the wet–dry treatment are taken into account using augmented Riemann solvers to build the Roe-based FOM. In addition to this, a time averaging approach is necessary to develop the Roe-based ROM. This approach is validated by solving some cases and the computed solutions are compared with those ones of Lax–Friedrichs-based ROMs. It is also studied whether the ROM preserves or not the well-balancing, the entropy fix and the wet–dry treatment