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Grid approximation of singularly perturbed boundary value problems with convective terms. (English) Zbl 0816.65051
Summary: The first boundary value problem is considered for a parabolic-type equation in an \(n\)-dimensional band domain. Highest-order derivatives of the equation contain a parameter taking arbitrary values in the half-open interval (0,1]. With the parameter value equal to zero, the equation degenerates into a transport equation which is a first-order equation containing derivatives with respect to spatial variables (this leads to an appearance of regular boundary layers).
The coefficients of the equation and the free term have discontinuities of the first kind on surfaces parallel to the sides of the band domain. On these surfaces there operate lumped sources. To solve the problem by using grids condensing in boundary and internal layers, a difference scheme is constructed which converges uniformly in the parameter everywhere in the domain.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35B25 Singular perturbations in context of PDEs