Advertisement

Numerical Algorithms

, Volume 53, Issue 4, pp 511–553 | Cite as

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations

  • Christina C. ChristaraEmail author
  • Tong Chen
  • Duy Minh Dang
Original Paper

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 American options 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Archer, D.: An O(h 4) cubic spline collocation method for quasilinear parabolic equations. SIAM J. Numer. Anal. 14, 620–637 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bialecki, B., Fairweather, G.: Orthogonal spline collocation methods for partial differential equations. J. Comput. Appl. Math. 128, 55–82 (2001). Numerical analysis 2000, vol. VII, Partial differential equationszbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, T.: An efficient algorithm based on quadratic spline collocation and finite difference methods for parabolic partial differential equations. Master’s thesis, University of Toronto, Toronto, Ontario, Canada (2005)Google Scholar
  4. 4.
    Christara, C.C.: Quadratic spline collocation methods for elliptic partial differential equations. BIT 34, 33–61 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Christara, C.C., Ng, K.S.: Adaptive techniques for spline collocation. Computing 76, 259–277 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Christara, C.C., Ng, K.S.: Optimal quadratic and cubic spline collocation on nonuniform partitions. Computing 76, 227–257 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dang, D.M.: Adaptive finite difference methods for valuing American options. Master’s thesis, University of Toronto, Toronto, Ontario (2007)Google Scholar
  8. 8.
    de Boor, C., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 582–606 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Douglas, J., Dupont, T.: A finite element collocation method for quasilinear parabolic equations. Math. Comput. 27, 17–28 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Douglas, J., Dupont, T.: Collocation methods for parabolic equations in a single-space variable. Lect. Notes Math. 385, 1–147 (1974)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Forsyth, P.A., Vetzal, K.: Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comput. 23, 2095–2122 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Greenwell-Yanik, C.E., Fairweather, G.: Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables. SIAM J. Numer. Anal. 23, 282–296 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Houstis, E.N., Christara, C.C., Rice, J.R.: Quadratic-spline collocation methods for two-point boundary value problems. Int. J. Numer. Methods Eng. 26, 935–952 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Houstis, E.N., Rice, J.R., Christara, C.C., Vavalis, E.A.: Performance of scientific software. In: Rice, J.R. (ed.) The IMA Volumes in Mathematics and its Applications. Mathematical Aspects of Scientific Software, vol. 14, pp. 123–155 (1988)Google Scholar
  15. 15.
    Hull, J.C.: Options, Futures, and Other Derivatives, 6th edn. Prentice Hall, Englewood Cliffs (2006)Google Scholar
  16. 16.
    Iserles, A.: A first Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1997)Google Scholar
  17. 17.
    Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. Math. 43, 309–327 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wang, R., Keast, P., Muir, P.: BACOL: B-Spline adaptive COLlocation software for 1-D parabolic PDEs. ACM Trans. Math. Softw. 30, 454–470 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Wang, R., Keast, P., Muir, P.: A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. Appl. Numer. Math. 50, 239–260 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Christina C. Christara
    • 1
    Email author
  • Tong Chen
    • 1
  • Duy Minh Dang
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations