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.     

Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε² multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge ε-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of ε in the prescribed bounded subdomains that can expand as the number of grid points increases. As ε → 0, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge ε-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of ε and the subdomains may expand as the number of nodes in the underlying grids increases.     

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ?², where ? ∈ (0, 1]. When ? is small, a boundary and an interior layer (with the characteristic width ?) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed ?, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + n ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in x and t, respectively. Based on the Richardson technique, a scheme that converges ?-uniformly at a rate of O(N ^?3 + N ₀ ^?2 ) is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in ?-uniformly in x with an order greater than three.     

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an ε-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an ε-uniform bound O(δ ₁ ^?2 lnδ ₁ ^?1 + δ ₀ ^?1 ) is obtained, where δ₁ and δ₀ are the error components due to the approximation of the derivatives with respect to x and t, respectively. Thus, this scheme is ε-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ); this scheme is not ε-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as ε → 0, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ). Neither the matrix of the ε-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is ε-uniformly well-conditioned.     

A method of asymptotic constructions with improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation

The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter ε that takes any values from the half-open interval (0, 1]. For this type of linear problems, the order of the ε-uniform convergence (with respect to x and t) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge ε-uniformly at the rate of O(N ^?2ln² N + N ^?2 ₀), where N + 1 and N ₀ + 1 are the numbers of the mesh points with respect to x and t, respectively. On the x axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions.” Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width O(ε ln N)) the first derivative with respect to x is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to x), the classical implicit difference schemes approximating the first derivative with respect to x by the central difference derivative are applied. To improve the accuracy in t, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter ε.     

Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters

In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the ith equation are multiplied by the perturbation parameter ? _i ² (i = 1, 2). The parameters ?_i take arbitrary values in the half-open interval (0, 1]. When the vector parameter ? = (?₁, ?₂) vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components ?₁ and (or) ?₂ tend to zero, a double boundary layer with the characteristic width ?₁ and ?₂ appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge ?-uniformly at the rate of O(N ^?2ln² N) are constructed, where N = min N _s, N _s + 1 is the number of mesh points on the axis x _s.     

Numerical solutions based on Chebyshev collocation method for singularly perturbed delay parabolic PDEs

The main purpose of this work is to provide a numerical approach for the delay partial differential equations based on a spectral collocation approach. In this research, a rigorous error analysis for the proposed method is provided. The effectiveness of this approach is illustrated by numerical experiments on two delay partial differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ?², where ? takes arbitrary values in the interval (0, 1]. When ? vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ? tends to zero, a parabolic boundary layer with a characteristic width ? appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 , where \(N = \mathop {\min }\limits_s N_s \), N ^s + 1 and N ₀ + 1 are the numbers of mesh points on the axes x _s and t, respectively.     

Grid approximation of singularly perturbed parabolic equations in the presence of weak and strong transient layers induced by a discontinuous right-hand side

The initial value problem on a line for singularly perturbed parabolic equations with convective terms is investigated. The first-and the second-order space derivatives are multiplied by the parameters ?₁ and ?₂, respectively, which may take arbitrarily small values. The right-hand side of the equations has a discontinuity of the first kind on the set $\bar \gamma $ = [x = 0] × [0, T]. Depending on the relation between the parameters, the appearing transient layers can be parabolic or regular, and the “intensity” of the layer (the maximum of the singular component) on the left and on the right of $\bar \gamma $ can be substantially different. If the parameter ?₂ at the convective term is finite, the transient layer is weak. For the initial value problems under consideration, the condensing grid method is used to construct finite difference schemes whose solutions converge (in the discrete maximum norm) to the exact solution uniformly with respect to ?₁ and ?₂ (when ?₂ is finite and, therefore, the transient layers are weak, no condensing grids are required).     

Parallel algorithms for a singularly perturbed parabolic problem

This article deals with iterative algorithms for domain decomposition applied to the solution of a singularly perturbed parabolic problem. These algorithms are based on finite difference domain decomposition methods and are suitable for parallel computing. Convergence properties of the algorithms are established. Numerical results for test problems are presented. © 1999 Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 389–405, 1999     

Concentrating solutions of some singularly perturbed elliptic equations

We study singularly perturbed elliptic equations arising from models in physics or biology, and investigate the asymptotic behavior of some special solutions. We also discuss some connections with problems arising in differential geometry.      

Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry

The boundary value problem for the singularly perturbed reaction-diffusion parabolic equation in a ball in the case of spherical symmetry is considered. The derivatives with respect to the radial variable appearing in the equation are written in divergent form. The third kind boundary condition, which admits the Dirichlet and Neumann conditions, is specified on the boundary of the domain. The Laplace operator in the differential equation involves a perturbation parameter ?², where ? takes arbitrary values in the half-open interval (0, 1]. When ? → 0, the solution of such a problem has a parabolic boundary layer in a neighborhood of the boundary. Using the integro-interpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in the radial and time variables, respectively.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

On the stability and domain of attraction of asymptotically nonsmooth stationary solutions to a singularly perturbed parabolic equation

A stationary solution to the singularly perturbed parabolic equation ?u _t + ε² u _xx ? f(u, x) = 0 with Neumann boundary conditions is considered. The limit of the solution as ε → 0 is a nonsmooth solution to the reduced equation f(u, x) = 0 that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found.     

An efficient hybrid numerical method based on an additive scheme for solving coupled systems of singularly perturbed linear parabolic problems

We construct an efficient hybrid numerical method for solving coupled systems of singularly perturbed linear parabolic problems of reaction-diffusion type. The discretization of the coupled system is based on the use of an additive or splitting scheme on a uniform mesh in time and a hybrid scheme on a layer-adapted mesh in space. It is proven that the developed numerical method is uniformly convergent of first order in time and third order in space. The purpose of the additive scheme is to decouple the components of the vector approximate solution at each time step and thus make the computation more efficient. The numerical results confirm the theoretical convergence result and illustrate the efficiency of the proposed strategy.     

Complete flux scheme for parabolic singularly perturbed differential‐difference equations

In this study, we investigate the concept of the complete flux (CF) obtained as a solution to a local boundary value problem (BVP) for a given parabolic singularly perturbed differential‐difference equation (SPDDE) with modified source term to propose an efficient complete flux‐finite volume method (CF‐FVM) for parabolic SPDDE which is μ‐ and ?‐uniform method where μ, ? are shift and perturbation parameters, respectively. The proposed numerical method is shown to be consistent, stable, and convergent and has been successfully implemented on three test problems.     

A higher order difference method for singularly perturbed parabolic partial differential equations

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.     

The Cauchy problem for a singularly perturbed Volterra integro-differential equation

The Cauchy problem for a singularly perturbed Volterra integro-differential equation is examined. Two cases are considered: (1) the reduced equation has an isolated solution, and (2) the reduced equation has intersecting solutions (the so-called case of exchange of stabilities). An asymptotic expansion of the solution to the Cauchy problem is constructed by the method of boundary functions. The results are justified by using the asymptotic method of differential inequalities, which is extended to a new class of problems.     

Convergence analysis of the LDG method applied to singularly perturbed problems

Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N^‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N²). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013