Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables. (English) Zbl 0595.65122

Greenwell-Yanik, C. E.; Fairweather, G.

SIAM J. Numer. Anal. 23, 282-296 (1986).

Optimal order error estimates are derived for the continuous-time collocation method and a discrete-time collocation method for approximating the solution of a semilinear parabolic initial-boundary value problem in \(R\times (0,T]\), R the unit square. At each time level, the approximation is a \(C^ 1\) piecewise polynomial of degree at least 3, determined 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 \(R\times (0,T]\).

Reviewer: Michael Sever (Jerusalem)

MSC:

65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations

Keywords:

spline collocation; Crank-Nicholson; Optimal order error estimates; collocation method; semilinear; Gauss points

Software:

PDECOL; COLSYS