Approximation of the solutions and diffusion flows of singularly perturbed boundary-value problems with discontinuous initial conditions

Boundary-value problems for a parabolic equation with mixed boundary conditions are considered on a strip. The highest derivatives of the equation contain a parameter which takes arbitrary values in the half-interval (0, 1]. If the parameter is equal to zero, the equation degenerates into a first-order equation which contains a derivative with respect to the time variable only. The initial condition has a discontinuity of the first kind. In problems of this kind, the error of the approximate solution, and the relative error of the calculated diffusion flows obtained from classical difference approximations of the boundary-value problem using uniform grids, increase without limit as the parameter tends to zero. The method of special clustering grids and the adjustment method are used to construct special difference schemes, by means of which the solution of the boundary-value problem and the diffusion flows can be approximated uniformly with respect to the parameter. (C) 1997 Elsevier Science Ltd.