Parameter-uniform error estimate for the implicit four-point scheme for a singularly perturbed heat equation with corner singularities

We study an initial-boundary value problem for a singularly perturbed one-dimensional heat equation on an interval. At the corner points, the input data are subjected to continuity conditions only, which violates the smoothness of the derivatives of the solution in neighborhoods of these points, starting from the derivatives occurring in the equation. To approximate the problem, we use the implicit four-point difference scheme on a Shishkin grid uniform with respect to time and piecewise uniform with respect to the space variable. We prove that the grid solution error is O(τ +N ^?2 ln² N) ln(j +1) uniformly with respect to the parameter, where τ is the grid increment with respect to the time variable, j is the index of the time layer, and N is the number of nodes in the piecewise uniform space grid.     

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.     

Convergence of the numerical solution of the initial-boundary value problem for a heat equation with a corner singularity in derivatives of the solution

An estimate O(τ + h ²)ln(j + 1) of the convergence rate for the solution of a four-point implicit difference scheme used for approximations on a uniform grid of a one-dimensional heat equation is obtained, with the provision that the boundary and initial data are subject at corner points only to the continuity condition, with no other compatibility conditions being satisfied. A discrete Green’s function is used to obtain an a priori estimate of the grid solution in terms of the appropriate negative norm of the right-hand side.     

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ?-uniformly well conditioned or ?-uniformly stable to perturbations of the data of the grid problem (here, ? is a perturbation parameter, ? ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ?-uniformly in the maximum norm at an O(N ^?1lnN + N ₀ ^?1 ) rate, where N + 1 and N ₀ + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ?-uniformly well conditioned and ?-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ^?2lnδ^?1 + δ ₀ ^?1 ); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N ^?1lnN and δ₀ = N ₀ ^?1 are the accuracies of the discrete solution in x and t, respectively.     

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 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.     

A Green’s function formulation of nonlocal finite-difference schemes for reaction-diffusion equations

Reaction-diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. The aim of this work is to show that a Green’s function formulation of reaction-diffusion PDEs is a suitable framework to derive FD schemes incorporating both O(h²) accuracy and nonlocal approximations in the whole domain (including boundary nodes). By doing so, the approach departs from a Green’s function formulation of the boundary-value problem to pose an approximation problem based on a domain decomposition. Within each subdomain, the corresponding integral equation is forced to have zero residual at given grid points. Different FD schemes are obtained depending on the numerical scheme used for computing the Green’s integral over each subdomain. Dirichlet and Neumann boundary conditions are considered, showing that the FD scheme based on the Green’s function formulation incorporates, in a natural way, the effects of boundary nodes in the discretization approximation.     

Solution of the Dirichlet problem for a singularly perturbed reaction-diffusion equation in a square on a Bakhvalov grid

The Dirichlet problem for a singularly perturbed reaction-diffusion equation in a square is solved with the help of the classic five-point difference scheme and a grid that is the tensor product of 1D Bakhvalov grids. Without imposing additional matching conditions in the corners of the domain, it is shown that the grid solution to the problem has the accuracy O(N ⁻²) in the norm L _∞ ^h , where N is the number of grid nodes along each direction. The accuracy is uniform with respect to a small parameter. A simulation confirms the theoretical prediction.     

Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ? (this is the coefficient of the higher order derivatives of the equation, ? ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S ₁ ^L . In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ?-uniformly at a rate of O(N ^?1lnN + N ₀), where N and N ₀ define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N ^?1 + N ₀ ^?1 ? ?. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S ₁ ^L with respect to x and t, the convergence under the condition N ^?1 + N ₀ ^?1 ≤ ?^1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ?-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ?-uniformly at a rate of O(N ^?1lnN + N ₀).     

A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation

For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ɛ, where ɛ ∈ (0, 1], the grid approximation of the Dirichlet problem on a rectangular domain in the (x, t)-plane is examined. For small ɛ, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of ɛ-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges ɛ-uniformly in the maximum norm at the rate of O(N ⁻²ln² N + N ₀⁻¹), where N + 1 and N ₀ + 1 are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme convergesɛ-uniformly at the rate of O(N ⁻⁴ln⁴ N + N ₀⁻²). For fixed values of the parameter, the convergence rate is O(N ⁻⁴ + N ₀⁻²).     

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain is considered for when the solution has singularities at the corners of the domain. The densification of the Shishkin mesh near the inner corner where different boundary conditions meet is such that the solution obtained by the classical five-point difference scheme converges to the solution of the initial problem in the mesh norm L _?? ^h uniformly with respect to the small parameter with almost second order, i.e., as a smooth solution. Numerical analysis confirms the theoretical result.     

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 mesh approximation to nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle

Here nonsmooth solutions of a differential equation are treated as solutions for which the compatibility conditions are not required to hold at the corner points of the domain and hence corner singularities can occur. In the present paper, we drop the compatibility conditions at three of the four vertices of a rectangle. At the remaining vertex, from which a characteristic (inclined) of the reduced equation issues, we impose compatibility conditions providing the C ^3,λ -smoothness of the desired solution in a neighborhood of that vertex as well as additional conditions leading to the smoothness of solutions of the reduced equation occurring in the regular component of the solution of the considered problem. Under our assumptions and for a sufficient smoothness of the coefficients of the equation and its right-hand side, we show that the classical five-point upwind approximation on a Shishkin piecewise uniform mesh preserves the accuracy specific for the smooth case; i.e., the mesh solution uniformly (with respect to a small parameter) converges in the L _∞ ^h -norm to the exact solution at the rate O(N ⁻¹ ln² N), where N is the number of mesh nodes in each of the coordinate directions.     

Computer difference scheme for a singularly perturbed elliptic convection–diffusion equation in the presence of perturbations

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε, ε ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge -uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if N ₁ ^-1 N ₂ ^-1 ? ε, where N ₁ and N ₂ are the numbers of grid intervals in x and y, respectively, the scheme is not -uniformly well-conditioned or ε-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on ε, N ₁,N ₂, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as N ₁,N ₂ → ∞, ε ∈ (0,1]. The difference schemes constructed in the presence of the indicated perturbations that converges as N ₁,N ₂ → ∞ for fixed ε, ε ∈ (0,1, is called a computer difference scheme. Schemes converging ε-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.     

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.     

Computer difference scheme for a singularly perturbed convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ? (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an \(\mathcal{O}\) (δ _st ) rate, where δ _st = (? + N ^?1)^?1 N ^?1 and N + 1 is the number of grid nodes; the scheme is not ?-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ? down to its complete loss for small ? (namely, for ? = \(\mathcal{O}\) (δ^?2max _i,j |δa _i ^j | + δ^?1 max _{i, j} |δb _i ^j |), where δ = δ _st and δa _i ^j , δb _i ^j are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ? ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Finite difference method in the problem of diffraction of a plane acoustic wave in a half-plane with a cut

In the numerical solution of the diffraction problem for an acoustic plane wave in a half-plane with a cut, boundary conditions that are equivalent to the radiation conditions at infinity are set in a neighborhood of the points of the cut. Joining the physical boundary conditions on the cut, a closing set of equations of order 4N, where N is the number of grid points on the cut, is obtained. The so-called Green’s grid function for the half-plane is used, which makes it possible to pass from one grid layer to another one for the solution satisfying certain conditions at infinity.     

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.     

A fourth order finite difference method for the Dirichlet biharmonic problem

Using the coupled approach, we formulate a fourth order finite difference scheme for the solution of the Dirichlet biharmonic problem on the unit square. On an N × N uniform partition of the square the scheme is solved at a cost O(N ² log₂ N)+m8N ² using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. Numerical tests confirm the fourth order accuracy of the scheme at the partition nodes with m proportional to log₂ N.