An accuracy-preserving numerical scheme for parabolic partial differential equations subject to discontinuous boundary conditions

In this paper, we develop a method to alleviate the loss of accuracy that occurs when parabolic partial differential equations (PDEs), subject to discontinuous boundary conditions, are solved numerically. Employing the Keller box finite-difference method, we consider a benchmark case involving the linear one-dimensional transient heat equation, subject to discontinuous heat flux at one of the boundaries. The method we develop constitutes an improvement on earlier work which involved dropping a spatial grid point at each time step; moreover, we demonstrate that our new approach is able to maintain second-order accuracy for the solution and the numerical scheme, whereas it is in general not even possible to calculate the accuracy for either using the earlier approach. Furthermore, our method can also be used in principle for discontinuities in non-linear steady-state boundary-layer problems, such as those that occur in fluid mechanics and electrochemistry; some examples of these are given.