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

Journal of Numerical Algorithms, July 2009

31 Pages Posted: 4 Oct 2009 Last revised: 7 Apr 2010

See all articles by Christina Christara

Christina Christara

University of Toronto - Department of Computer Science

Tong Chen

University of Toronto - Department of Computer Science

Duy-Minh Dang

University of Queensland - School of Mathematics and Physics

Date Written: July 2, 2009

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

Suggested Citation

Christara, Christina and Chen, Tong and Dang, Duy-Minh, Quadratic Spline Collocation for One-Dimensional Linear Parabolic Partial Differential Equations (July 2, 2009). Journal of Numerical Algorithms, July 2009. Available at SSRN: https://ssrn.com/abstract=1482071

Christina Christara

University of Toronto - Department of Computer Science ( email )

Department of Computer Science
University of Toronto
Toronto, Ontario M5S 3G4
Canada

Tong Chen

University of Toronto - Department of Computer Science ( email )

Sandford Fleming Building
10 King's College Road, Room 3302
Toronto, Ontario M5S 3G4
Canada

Duy-Minh Dang (Contact Author)

University of Queensland - School of Mathematics and Physics ( email )

Priestly Building
St Lucia
Brisbane, Queesland 4067
Australia

HOME PAGE: http://people.smp.uq.edu.au/Duy-MinhDang/

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