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Open Access
January 1, 2003
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Open Access
January 1, 2003
### Abstract

A class of singularly perturbed time-dependent convection-diffusion problems with a boundary turning point is examined on a rectangular domain. The solution of problems from this class possesses a parabolic boundary layer in the neighborhood of one of the sides of the domain. Classical numerical methods on uniform meshes are known to be inadequate for problems with boundary layers. A numerical method consisting of a standard upwind finite difference operator on a fitted mesh is constructed. It is proved that the numerical approximations generated by this method converge uniformly with respect to the singular perturbation parameter. Numerical results are presented that verify computationally the theoretical result.

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Open Access
January 1, 2003
### Abstract

We consider the evolution of a thin film of viscous fluid on the inside surface of a cylinder with the horizontal axis, rotating with a constant angular velocity about this axis. We use a lubrication approximation extended to the first order in the dimensionless film thickness (including the small effects of the variation of the film pressure across its thickness and the surface tension) and numerically we compute the time evolution of the film to a steady state.

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Open Access
January 1, 2003
### Abstract

New high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε. Solutions of the well-known classical numerical schemes for such problems do not converge ε-uniformly (the errors of such schemes depend on the value of the parameter ε and are comparable with the solution itself for small values of ε). The convergence order of the existing ε-uniformly convergent schemes does not exceed 1 in space and time. In this paper, using a defect correction technique, we construct a special difference scheme that converges ε-uniformly with the second (up to a logarithmic factor) order of accuracy with respect to x and with the second order of accuracy and higher with respect to t.

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Open Access
January 1, 2003
### Abstract

The flow of a viscous incompressible uid past a body of revolution with aparabolic profile when the stream is parallel to its axis falls into a class of problems that exhibit boundary layers. This problem does not have solutions in closed form, and is modeled by boundary-layer equations. Using a self-similar approach based on a Blasius series expansion (up to two terms), the boundary-layer equations can be reduced to a Blasius-type problem consisting of a system of three 3rd order ordinary differential equations on a semi-infinite interval. Numerical methods need to be employed to attain the solutions of these equations and their derivatives, which are required for the computation of the velocity components, on a finite domain with accuracy independent of the viscosity v, which can take arbitrary values from the interval (0; 1]. Numerical methods for which the accuracy of the velocity components depend on the number of mesh points N, used to solve the Blasius-type problem, and do not depend on the viscosity v, are referred to as robust methods. To construct a robust numerical method we reduce the original problem on a semi-infinite axis to a problem on the finite interval [0;K], where K = K(N) = lnN. Employing numerical experiments, we justify that the constructed numerical method is parameter robust.

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Open Access
January 1, 2003
### Abstract

We consider a central difference scheme for the numerical solution of a system of coupled reaction-diffusion equations. We show that the scheme is almost second-order convergent, uniformly in the perturbation parameter. We present the results of numerical experiments to confirm our theoretical results.

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Open Access
January 1, 2003
### Abstract

In this paper, parameter - uniform numerical methods for singularly perturbed ordinary differential equations containing two small parameters are studied.Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of numerical approximations.

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Open Access
January 1, 2003
### Abstract

A singularly perturbed convection-diffusion problem with two small parameters is considered. The problem is solved using the streamline-diffusion finite element method on a Shishkin mesh. We prove that the method is convergent independently of the perturbation parameters. Numerical experiments support these theoretical results.

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Open Access
January 1, 2003
### Abstract

In this paper we discuss conditioning of a discrete Schwarz method on piecewise–uniform meshes with an example of a one-dimensional singularly perturbed boundary-value problem. We consider a Dirichlet problem for singularly perturbed ordinary differential equations with convection terms and a small perturbation parameter ε. To solve the problem numerically we use an ε-uniformly convergent (in the maximum norm) difference scheme on special piecewise–uniform meshes. For this base scheme we construct a decomposition scheme based on a Schwarz technique with overlapping subdomains, which converges ε-uniformly with respect to both the number of mesh points and the number of iterations. The step-size of such special meshes is extremely small in the neighborhood of the layer and changes sharply on its boundary, that (as was shown by A.A. Samarskii) can generally lead to a loss of well-conditioning of the above schemes. For the decomposition scheme we study the conditioning of the system (difference scheme) and the conditioning of the system matrix (difference operator), and also the influence of perturbations in the data of the boundary-value problem on disturbances of its numerical solutions. We derive estimates for the disturbances of the numerical solutions (in the maximum norm) depending on the subdomain in which the disturbance of the data appears. It is shown that the condition number of the difference operator associated with the Schwarz method, just as for the base scheme, is not ε-uniformly bounded. However, these difference schemes are well-conditioned ε-uniformly (with the ε-uniform estimate for the condition number being the same as for the schemes on uniform meshes for regular problems) when the right-hand side of the discrete equations is considered in a “natural” norm, i.e., in the maximum norm with a special weight multiplier. In the case of the boundary-value problem with perturbed data we give conditions under which the solution of the iterative scheme based on the overlapping Schwarz method is convergent ε-uniformly to the solution of this Dirichlet problem as the number of mesh points and the number of iterations increase.

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Open Access
January 1, 2003
### Abstract

A singularly perturbed two-point boundary-value problem is considered. Working in the discrete maximum norm, a necessary condition for the convergence (uniformly in the singular perturbation parameter) of general difference schemes on general meshes is proved. This encompasses both a 1976 result of Miller for uniform meshes and more recent results of the same author that deal with piecewise uniform Shishkin meshes.

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Open Access
January 1, 2003
### Abstract

This paper presents a convergence analysis for the exponentially fitted finite volume method in two dimensions applied to a linear singularly perturbed convection-diffusion equation with exponential boundary layers. The method is formulated as a nonconforming Petrov-Galerkin finite element method with an exponentially fitted trial space and a piecewise constant test space. The corresponding bilinear form is proved to be coercive with respect to a discrete energy norm. Numerical results are presented to verify the theoretical rates of convergence.