An orthogonal spline collocation alternating direction implicit Crank-Nicolson method for linear parabolic problems on rectangles

We formulate an alternating direction implicit Crank-Nicolson scheme for solving a general linear variable coefficient parabolic problem in nondivergence form on a rectangle with the solution subject to nonhomogeneous Dirichlet boundary condition. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. We show that for sufficiently small time stepsize the scheme is stable and of optimal-order accuracy in time and the H¹ norm in space. We also describe an efficient implementation of the scheme and present numerical results demonstrating the accuracy and convergence rates in various norms.