A singular perturbation approach to solve Burgers–Huxley equation via monotone finite difference scheme on layer-adaptive mesh

This paper deals with a numerical method for solving one-dimensional unsteady Burgers–Huxley equation with the viscosity coefficient ε. The parameter ε takes any values from the half open interval (0, 1]. At small values of the parameter ε, an outflow boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem with a singular perturbation parameter ε. Using singular perturbation analysis, asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. We construct a numerical scheme that comprises of implicit-Euler method to discretize in temporal direction on uniform mesh and a monotone hybrid finite difference operator to discretize the spatial variable with piecewise uniform Shishkin mesh. To obtain better accuracy, we use central finite difference scheme in the boundary layer region. Shishkin meshes are refined in the boundary layer region, therefore stability constraint is satisfied by proposed scheme. Quasilinearization process is used to tackle the non-linearity and it is shown that quasilinearization process converges quadratically. The method has been shown to be first order uniformly accurate in the temporal variable, and in the spatial direction it is first order parameter uniform convergent in the outside region of boundary layer, and almost second order parameter uniform convergent in the boundary layer region. Accuracy and uniform convergence of the proposed method is demonstrated by numerical examples and comparison of numerical results made with the other existing methods.     

High order parameter-robust numerical method for singularly perturbed reaction-diffusion problems

We present a high order parameter-robust finite difference method for singularly perturbed reaction-diffusion problems. The problem is discretized using a suitable combination of fourth order compact difference scheme and central difference scheme on generalized Shishkin mesh. The convergence analysis is given and the method is proved to be almost fourth order uniformly convergent in maximum norm with respect to singular perturbation parameter ε. Numerical experiments are conducted to demonstrate the theoretical results.     

Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014     

A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems

This paper deals with the study on system of reaction diffusion differential equations for Robin or mixed type boundary value problems (MBVPs). A cubic spline approximation has been used to obtain the difference scheme for the system of MBVPs, on a piecewise uniform Shishkin mesh defined in the whole domain. It has been shown that our proposed scheme, i.e., central difference approximation for outer region with cubic spline approximation for inner region of boundary layers, leads to almost second order parameter uniform convergence whereas the standard method i.e., the forward-backward approximation for mixed boundary conditions with central difference approximation inside the domain leads to almost first order convergence on Shishkin mesh. Numerical results are provided to show the efficiency and accuracy of these methods.     

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.     

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter ε, the size of the uniform mesh in x, the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on ε. The scheme converges almost ε-uniformly; namely, it converges under the condition N ^?1 = o(ε^v), where v = v(K) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final Kth iteration, the difference scheme converges ε-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known ε-uniformly convergent schemes on piecewise uniform grids.     

A coupled continuous-discontinuous FEM approach for convection diffusion equations

In this article, we introduce a coupled approach of local discontinuous Galerkin and standard finite element method for solving convection diffusion problems. The whole domain is divided into two disjoint subdomains. The discontinuous Galerkin method is adopted in the subdomain where the solution varies rapidly, while the standard finite element method is used in the other subdomain due to its lower computational cost. The stability and a priori error estimate are established. We prove that the coupled method has O((ε1 / 2 + h 1 / 2 )h k ) convergence rate in an associated norm, where ε is the diffusion coefficient, h is the mesh size and k is the degree of polynomial. The numerical results verify our theoretical results. Moreover, 2k-order superconvergence of the numerical traces at the nodes, and the optimal convergence of the errors under L 2 norm are observed numerically on the uniform mesh. The numerical results also indicate that the coupled method has the same convergence order and almost the same errors as the purely LDG method.     

A two-grid method for elliptic problem with boundary layers

A two-grid method for the elliptic equation with a small parameter ε multiplying the highest derivative is investigated. The difference schemes with the property of ε-uniform convergence on a uniform mesh and on Shishkin mesh are considered. In both cases, a two-grid method for resolving the difference scheme is investigated. A two-grid method has features that are concerned with a uniform convergence of a difference scheme. To increase the accuracy, the Richardson extrapolation in two-grid method is applied. Numerical results are discussed.     

Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform B-spline collocation method

This paper is concerned with a numerical scheme to solve a singularly perturbed convection-diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects.     

A C0 interior penalty method for a singularly‐perturbed fourth‐order elliptic problem on a layer‐adapted mesh

We analyze the convergence of a continuous interior penalty (CIP) method for a singularly perturbed fourth‐order elliptic problem on a layer‐adapted mesh. On this anisotropic mesh, we prove under reasonable assumptions uniform convergence of almost order k ? 1 for finite elements of degree k ≥ 2. This result is of better order than the known robust result on standard meshes. A by‐product of our analysis is an analytic lower bound for the penalty of the symmetric CIP method. Finally, our convergence result is verified numerically. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 838–861, 2014     

Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation

The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1?d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes to zero. Then we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. Our method of proof relies essentially on discrete multiplier techniques.     

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.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

A higher order finite element method with modified graded mesh for singularly perturbed two-parameter problems

This paper presents a modified graded mesh for singularly perturbed two-parameter problems. The mesh is generated recursively using Newton's algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on higher order polynomials of degree p≥1. We prove parameter uniform convergence of optimal order in ε-weighted energy norm. A test example is taken to compare the proposed graded mesh with others found in the literature.     

Sequential and parallel domain decomposition methods for a singularly perturbed parabolic convection-diffusion equation

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered in a rectangular domain in x and t; the perturbation parameter ? multiplying the highest derivative takes arbitrary values in the half-open interval (0,1]. For the boundary value problem, we construct a scheme based on the method of lines in x passing through N ₀+1 points of the mesh with respect to t. To solve the problem on a set of intervals, we apply a domain decomposition method (on overlapping subdomains with the overlap width δ), which is a modification of the Schwarz method. For the continual schemes of the decomposition method, we study how sequential and parallel computations, the order of priority in which the subproblems are sequentially solved on the subdomains, and the value of the parameter ? (as well as the values of N ₀, δ) influence the convergence rate of the decomposition scheme (as N ₀ → ∞), and also computational costs for solving the scheme and time required for its solution (unless a prescribed tolerance is achieved). For convection-diffusion equations, in contrast to reaction-diffusion ones, the sequential scheme turns out to be more efficient than the parallel scheme.     

Higher order global solution and normalized flux for singularly perturbed reaction-diffusion problems

In this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction-diffusion two-point boundary-value problems. The HOC (higher order compact) finite difference scheme developed in Gracia et al. (2001) [4] and which is constructed on an appropriate piecewise uniform Shishkin mesh, has been considered to find an almost fourth order convergent solution at mesh points. Using these values, piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined. It has been shown that the global solution and the global normalized flux are also uniformly convergent. Moreover, for the global solution, the order of uniform convergence in the whole domain is optimal, i.e., it is the same as this one obtained at mesh points, whereas, for the global normalized flux, the uniform convergence is almost third order, except at midpoints of the mesh, where it is also almost fourth order. Theoretical error bounds have been provided along with some numerical examples, which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux.     

Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system

We prove a stability result of constant equilibria for the three dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter ε while the incompressible part of the initial velocity is assumed to be small compared to ε. We then get a unique global smooth solution. We also prove a uniform in ε time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at ε fixed and the dispersive techniques (dispersive estimates and normal forms) that are useful for the inviscid irrotational system.     

Robust convergence of a compact fourth-order finite difference scheme for reaction–diffusion problems

We consider a singularly perturbed one-dimensional reaction–diffusion problem with strong layers. The problem is discretized using a compact fourth order finite difference scheme. Altough the discretization is not inverse monotone we are able to establish its maximum-norm stability and to prove its pointwise convergence on a Shishkin mesh. The convergence is uniform with respect to the perturbation parameter. Numerical experiments complement our theoretical results.     

Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a small parameter ? (? ?? (0, 1]) multiplying the higher order derivative is considered. For the problem, a difference scheme on locally uniform meshes is constructed that converges in the maximum norm conditionally, i.e., depending on the relation between the parameter ? and the value N defining the number of nodes in the mesh used; in particular, the scheme converges almost ?-uniformly (i.e., its accuracy depends weakly on ?). The stability of the scheme with respect to perturbations in the data and its conditioning are analyzed. The scheme is constructed using classical monotone approximations of the boundary value problem on a priori adapted grids, which are uniform on subdomains where the solution is improved. The boundaries of these subdomains are determined by a majorant of the singular component of the discrete solution. On locally uniform meshes, the difference scheme converges at a rate of O(min[?^?1 N ^?K lnN, 1] + N ^?1lnN), where K is a prescribed number of iterations for refining the discrete solution. The scheme converges almost ?-uniformly at a rate of O(N ^?1lnN) if N ^?1 ?? ?^??, where ?? (the defect of ?-uniform convergence) determines the required number K of iterations (K = K(??) ?? ??^?1) and can be chosen arbitrarily small from the half-open interval (0, 1]. The condition number of the difference scheme satisfies the bound ?? _P = O(?^?1/K ln^1/K ?^?1??^{?(K + 1)/K} ), where ?? is the accuracy of the solution of the scheme in the maximum norm in the absence of perturbations. For sufficiently large K, the scheme is almost ?-uniformly strongly stable.