Quadratic Spline Collocation for One-Dimensional Linear Parabolic Partial Differential Equations

Abstract

New methods for solving general linear parabolic partial differential equations (PDEs) in one space dimension are developed. The methods combine quadratic-spline collocation for the space discretization and classical finite differences, such as Crank-Nicolson, for the time discretization. The main computational requirements of the most efficient method are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth order. The stability and convergence properties of some of the new methods are analyzed for a model problem. Numerical results demonstrate the stability and accuracy of the methods. Adaptive mesh techniques are introduced in the space dimension, and the resulting method is applied to the American put option pricing problem, giving very competitive results.

Keywords: Quadratic splines, Collocation, Parabolic PDEs, Crank-Nicolson, Stability, Optimal order of convergence, Adaptivity, American options

JEL Classification: G12, G13, C61, C63