Numerical Simulation and Applications of the Convection–Diffusion–Reaction Equation with the Radial Basis Function in a Finite-Difference Mode

42 Pages Posted: 13 May 2020

See all articles by Reza Mollapourasl

Reza Mollapourasl

Farmingdale State College,SUNY

Majid Haghi

Shahid Rajaee Teacher Training University

Alfa Heryudono

University of Massachusetts Dartmouth

Date Written: May 11, 2020

Abstract

This paper develops two local mesh-free methods for designing stencil weights and spatial discretization, respectively, for parabolic partial differential equations (PDEs) of convection–diffusion–reaction type. These are known as the radial-basis-function generated finite-difference method and the Hermite finite-difference method. The convergence and stability of these schemes are investigated numerically using some examples in two and three dimensions with regularly and irregularly shaped domains. Then we consider the numerical pricing of European and American options under the Heston stochastic volatility model. The European option leads to the solution of a two-dimensional parabolic PDE, and the price of the American option is given by a linear complementarity problem with a two-dimensional parabolic PDE of convection–diffusion–reaction type. Then we use the operator splitting method to perform time-stepping after space discretization. The resulting linear systems of equations are well conditioned and sparse, and by numerical experiments we show that our numerical technique is fast and stable with respect to the change in the shape parameter of the radial basis function. Finally, numerical results are provided to illustrate the quality of approximation and to show how well our approach converges with the results presented in the literature.

Keywords: convection–diffusion–reaction equation, radial basis functions (RBFs), Hermite finite difference, option pricing, stochastic volatility, Heston equation

Suggested Citation

Mollapourasl, Reza and Haghi, Majid and Heryudono, Alfa, Numerical Simulation and Applications of the Convection–Diffusion–Reaction Equation with the Radial Basis Function in a Finite-Difference Mode (May 11, 2020). Journal of Computational Finance, Vol. 23, No. 5, 2020, Available at SSRN: https://ssrn.com/abstract=3598016

Reza Mollapourasl (Contact Author)

Farmingdale State College,SUNY ( email )

School of Busines
Farmingdale, NY 11735

Majid Haghi

Shahid Rajaee Teacher Training University ( email )

Iran

Alfa Heryudono

University of Massachusetts Dartmouth ( email )

285 Old Westport Road
N Dartmouth, MA 02747
United States

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