Orthogonal spline collocation methods for partial differential equations

This paper provides an overview of the formulation, analysis and implementation of orthogonal spline collocation (OSC), also known as spline collocation at Gauss points, for the numerical solution of partial differential equations in two space variables. Advances in the OSC theory for elliptic boundary value problems are discussed, and direct and iterative methods for the solution of the OSC equations examined. The use of OSC methods in the solution of initial–boundary value problems for parabolic, hyperbolic and Schrödinger-type systems is described, with emphasis on alternating direction implicit methods. The OSC solution of parabolic and hyperbolic partial integro-differential equations is also mentioned. Finally, recent applications of a second spline collocation method, modified spline collocation, are outlined.