The alternating direction implicit (ADI) method
is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on
When the ADI technique is coupled with orthogonal spline collocation
(OSC) for discretization in space we not only obtain the global
solution efficiently but the discretization error with respect to space
variables can be of an arbitrarily high order. In ,
we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic
problems with Robin's boundary conditions on rectangular
polygons and demonstrated numerically the accuracy in various norms.
question that arises is: Does this method have an extension to non-rectangular regions? In this paper, we present a simple idea of how the
ADI OSC technique can be extended to some such regions. Our approach depends on the
transfer of Dirichlet boundary conditions in the solution of
a two-point boundary value problem (TPBVP).
We illustrate our idea for the solution of the heat equation on the unit disc
using piecewise Hermite cubics.