Efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles

Piecewise Hermite bicubic orthogonal spline collocation Laplace-modified and alternating-direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed. The schemes are shown to be unconditionally stable and of optimal order accuracy in the H¹ and discrete maximum norms for space and time, respectively. Implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms.