Analyses of Spline Collocation Methods for Parabolic and Hyperbolic Problems in Two Space Variables

Abstract

Optimal order error estimates are derived for the continuous-time collocation method and a discrete-time collocation method (the Crank Nicolson collocation method) for approximating the solution of a semilinear parabolic initial-boundary value problem in $R \times ( {0,T} ]$, where R is the unit square. In each case, at each time level, the approximation is a $C^1 $ piecewise polynomial of degree $r \geqq 3$ defined by collocation at Gauss points in R. Optimal order error estimates are also derived for a family of discrete-time collocation methods for a semilinear second order hyperbolic initial-boundary value problem in $R \times ( {0,T} ]$.