ε-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems

In this paper we study the discrete approximation of a Dirichlet problem on an interval for a singularly perturbed parabolic PDE. The highest derivative in the equation is multiplied by an arbitrarily small parameter e. If the parameter vanishes, the parabolic equation degenerates 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. Then the error in the approximate solution depends on the value of e. An adapted placement of the nodes is needed to ensure that the error independent of the parameter value and depends only on the number of nodes in the mesh. Special schemes with this property are called e-uniformly convergent. In this paper such e-uniformly convergent schemes are studied, which combine a difference scheme and a mesh selection criterion for the space discretization. Except for a small logarithmic factor, the order of convergence is one and two with respect to the time and space variables, respectively. Therefore, it is of interest to develop methods for which the order of convergence with respect to the time variable is increased. In this paper we develop schemes for which the order of convergence in time can be arbitrarily large if the solution is sufficiently smooth. To obtain uniform convergence, we use a mesh with nodes that are condensed in the neighbourhood of the boundary layers. To obtain a better approximation in time, we use auxiliary discrete problems on the same time-mesh to correct the difference approximations. In this sense, the present algorithm is an improvement over a previously published one. To validate the theoretical results, some numerical results for the new schemes are presented.