The optimization of methods of solving boundary value problems with a boundary layer

Abstract

THE solutions of a number of value problems for differential equations with small parameters having higher derivatives possess singularities of the boundary layer type [1, 2]. For the solution of such problems by finite-difference methods the integration step near the boundary must be substantially less than the thickness of the boundary layer which is a characteristic dimension of the problem. In the case of constant steps throughout the whole region of integration this circumstance leads to a considerable increase in the volume of calculations when the parameters are reduced with higher derivatives. An exception may be only the so-called “quasiclassical approximations” of [3], adapted specially for the solution of problems with small parameters having higher derivatives, but these can only be written for isolated classes of ordinary differential equations. The use of asymptotic methods of solution [2, 4] requires these parameters to be rather small, and their coefficients to be very smooth. The same often applies to the use of numerical methods. Below we construct, for two model boundary value problems (a set of ordinary differential equations and an elliptic equation), methods of solution on a network with variable steps and an evaluation of the error which is homogeneous in the small parameter, but with few constraints on the smoothness of the coefficients.