An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only O(N ) operations where N is the number of unknowns. Moreover, it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties.

Abstract. A complete stability and convergence analysis is given for two-and three-level, piecewise Hermite bicubic orthogonal spline collocation, Laplacemodified and alternating-direction schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimal-order accuracy in space and time.

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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 H 1 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.

A new alternating-direction implicit (ADI) Galerkin method is devised and analyzed for solving a certain class of second-order hyperbolic initial-boundary value problems in two space variables. This class includes the wave equation in Cartesian coordinates, polar coordinates, and cylindrical coordinates with radial symmetry. Optimal a priori H-and L2-error estimates are derived.