An Alternating-Direction Implicit Orthogonal Spline Collocation Scheme for Nonlinear Parabolic Problems on Rectangular Polygons

Abstract

We consider a nonlinear parabolic initial-boundary value problem on a rectangular polygon with the solution satisfying Robin boundary conditions with variable coefficients. An approximation to the solution at the desired time value is obtained using an alternating-direction implicit extrapolated Crank--Nicolson scheme in which orthogonal spline collocation with piecewise polynomials of an arbitrary degree is used for spatial discretization. At each time level, the scheme determines the intermediate solution along horizontal lines and the approximate solution along vertical lines passing through the Gauss points. Only at the last time level is the approximate solution along vertical lines converted into the approximate solution defined on the entire rectangular polygon. This property of our approach leads to its efficient implementation and its applicability to rectangular polygons.