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.     

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.     

On error estimates for Galerkin spectral discretizations of parabolic problems with nonsmooth initial data

We analyze the Legendre and Chebyshev spectral Galerkin semidiscretizations of a one dimensional homogeneous parabolic problem with nonconstant coefficients. We present error estimates for both smooth and nonsmooth data. In the Chebyshev case a limit in the order of approximation is established. On the contrary, in the Legendre case we find an arbitrary high order of convegence.

     

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

The objective of this paper is to construct and analyze a fitted operator finite difference method (FOFDM) for the family of time‐dependent singularly perturbed parabolic convection–diffusion problems. The solution to the problems we consider exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then, we discretize the time variable using the classical Euler method. This results in a system of singularly perturbed interior layer two‐point boundary value problems. We propose a FOFDM to solve the system above. Through a rigorous error analysis, we show that the scheme is uniformly convergent of order one with respect to both time and space variables. Moreover, we apply Richardson extrapolation to enhance the accuracy and the order of convergence of the proposed scheme. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme.     

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     

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.     

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.     

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.     

Two-point boundary value problem for a singularly perturbed equation with a reduced equation having multiple roots

Contrast steplike structures for a singularly perturbed equation are considered in the case when the corresponding reduced equation has multiple roots.     

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.     

On the structure of the solution of singularly perturbed initial boundary value problems with an unbounded spectrum of the limit operator

Singularly perturbed initial boundary value problems are studied for some classes of linear systems of ordinary differential equations on the semiaxis with an unbounded spectrum of the limit operator. We give a new version of the proof of the existence of a unique and bounded (as ε→+0) solution for which with the help of the splitting method we construct a uniform asymptotic expansion on the entire semiaxis and describe all singularities (reflecting the structure of the corresponding boundary layers) in closed analytic form, including the critical case in which the points of the spectrum of the limit operator can touch the imaginary axis; this supplements previous results. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 831–835, June, 1999.     

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.     

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.     

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter ? taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost ?-uniformly, i.e., with an error that weakly depends on the parameter ?: |u(x, t) ? z(x, t)| ≤ M[N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀ + ?^?1 N ₁ ^?K ln ^K?1 N ₁], (x, t) ε ? _h , where N ₁ + 1 and N ₀ + 1 are the numbers of grid points in x and t, respectively; K is the number of refinement iterations (with respect to x) in the adapted grid; and M = M(K). Outside the σ-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges ?-uniformly at a rate O(N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀), where σ ≤ MN ₁ ^{?K + 1} ln ^K?1 N ₁ for K ≥ 2.     

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).     

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.     

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.     

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.     

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.