The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

We present new results in the numerical analysis of singularly perturbed convection-diffusion-reaction problems that have appeared in the last five years. Mainly discussing layer-adapted meshes, we present also a survey on stabilization methods, adaptive methods, and on systems of singularly perturbed equations.

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Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

This paper is concerned with the fuzzy-model-based nonfragile control problem for discrete-time nonlinear singularly perturbed systems with stochastic jumping parameters. The stochastic parameters are generated from the semi-Markov process. The memory property of the transition probabilities among subsystems is fully considered in the investigated systems. Consequently, the restriction that the transition probabilities are memoryless in widely used discrete-time Markov jump model can be removed. Based on the T-S fuzzy model approach and semi-Markov kernel concept, several criteria ensuring $\delta$ -error mean square stability of the underlying closed-loop system are established. With the help of those criteria, the designed procedures which could well deal with the fragility problem in the implementation of the proposed fuzzy-model-based controller are presented. A technique is developed to estimate the permissible maximum value of singularly perturbed parameter for discrete-time nonlinear semi-Markov jump singularly perturbed systems. Finally, the validity of the established theoretical results is illustrated by a numerical example and a modified tunnel diode circuit model.

Fuzzy-Model-Based Nonfragile Control for Nonlinear Singularly Perturbed Systems With Semi-Markov Jump Parameters

A brief survey on numerical methods for solving singularly perturbed problems

A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.

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A parameter robust numerical method for a two dimensional reaction-diffusion problem

In this work we construct and analyze some finite difference schemes used to solve a class of time-dependent one-dimensional convection-diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank-Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples which illustrate the theoretical results, in the case of using the Euler implicit method, and give better numerical behaviour than that predicted theoretically, showing order two in time and order N−2log2N in space, if the Crank-Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A-stable SDIRK with two stages and a third-order HODIE difference scheme, showing its uniformly convergent behavior, reaching order three, up to a logarithmic factor. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

In this paper we develop a numerical method for two-dimensional time-dependent reaction-diffusion problems. This method, which can immediately be generalized to higher dimensions, is shown to be uniformly convergent with respect to the diffusion parameter.

C. Clavero

Papers

A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems

A parameter robust numerical method for a two dimensional reaction-diffusion problem

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problems

A fractional step method on a special mesh for the resolution multidimensional evolutionary convection-diffusion problems