New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) methods based on the backward Euler, the Crank–Nicolson, and the second order BDF methods combined with judiciously chosen quadrature rules are considered. The ADI OSC methods are proved to be of optimal accuracy in time and in the $L^2$ norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the $L^{\infty}$ and $H^1$norms and superconvergence phenomena.

Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the θ-method, for θ ∈ [1/2, 1], (where θ = 1 yields the backward Euler method and θ = 1/2 yields the Crank–Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.

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_0^1 $ and $L^2 $-error estimates are derived.

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.