In this paper the authors consider the discrete approximation of a Dirichlet problem on an interval for a singularly perturbed parabolic PDE. As is the case in such problems the highest derivative in the equation is multiplied by an arbitrarily small parameter ε. If the parameter goes to zero, the parabolic equation reduces to a first-order equation, in which only the time derivative remains. For small values of the parameter, boundary layers may appear that give rise to difficulties when classical discretization methods are applied. The error in the approximate solution depends on the value of ε. A special mesh is required to ensure that the error is independent of the parameter value and depends only on the number of mesh points. Such meshes are referred to as ε-uniformly convergent schemes. In this paper such schemes are studied, which combine a difference scheme and a mesh selection criterion for the space discretization.

The paper focuses on methods for which the order of convergence with respect to the time variable can be arbitrarily large if the solution is sufficiently smooth. To obtain uniform convergence, Shishkin meshes are used with nodes that are condensed in the neighbourhood of the boundary layers. To obtain a better approximation in time, auxiliary discrete problems on the same time-mesh to correct the difference approximations. In this sense, the algorithm presented is an improvement over a previously published one. To validate the theoretical results, some numerical results for the new schemes are presented.