Journal of Scientific Computing

, Volume 64, Issue 2, pp 341–367 | Cite as

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

  • Ali Safdari-Vaighani
  • Alfa Heryudono
  • Elisabeth Larsson
Article

Abstract

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF–PUM) proposed here. The objective of this paper is to establish that RBF–PUM is viable for parabolic PDEs of convection–diffusion type. The stability and accuracy of RBF–PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection–diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF–PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF–PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

Keywords

Collocation method Meshfree Radial basis function Partition of unity RBF–PUM Convection–diffusion equation American option 

Mathematics Subject Classification

MSC 65M70 MSC 35K15 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Ali Safdari-Vaighani
    • 1
  • Alfa Heryudono
    • 2
  • Elisabeth Larsson
    • 3
  1. 1.Department of MathematicsAllameh Tabataba’i UniversityTehranIran
  2. 2.Department of MathematicsUniversity of Massachusetts DartmouthDartmouthUSA
  3. 3.Department of Information TechnologyUppsala UniversityUppsalaSweden

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