The Crank–Nicolson orthogonal spline collocation method for one-dimensional parabolic problems with interfaces

One-dimensional parabolic problems with interfaces are solved using a method in which orthogonal spline collocation (OSC) is employed for the spatial discretization and the Crank–Nicolson method for the time-stepping. The derivation of the method is described in detail for the case in which cubic monomial basis functions are used in the development of the OSC discretization. With this background, the method is easily extended to monomials of higher degree. No matter the degree of the monomials, the method requires the solution of an almost block diagonal linear system at each time step using an existing algorithm at a cost of O(N2) on a spatial partition of N subintervals. The results of extensive numerical experiments involving examples from the literature are presented. Both cubic and quartic basis functions are employed. In each case, the results exhibit optimal global accuracy in the L∞,L2 and H1 spatial norms and second order accuracy in time. Moreover, superconvergence is observed at the nodes.