Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions (Advances in Applied Mathematics and Mechanics, 2021)
The alternating direction implicit (ADI) methodis a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems onrectangles.When the ADI technique is coupled with orthogonal spline collocation(OSC) for discretization in space, we not only obtain the globalsolution efficiently, but the discretization error with respect to spacevariables can be of an arbitrarily high order. In [2],we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolicproblems with Robin's boundary conditions on rectangularpolygons and demonstrated numerically the accuracy in various norms.A naturalquestion that arises is: Does this method have an extension to non-rectangular regions? In this paper, we present a simple idea of how theADI OSC technique can be extended to some such regions. Our approach depends on thetransfer of Dirichlet boundary conditions in the solution ofa two-point boundary value problem (TPBVP).We illustrate our idea for the solution of the heat equation on the unit discusing piecewise Hermite cubics.