Fourth order compact scheme for space fractional advection–diffusion reaction equations with variable coefficients

Abstract

In this work, a new fourth order compact approximation is derived for Riemann–Liouville space fractional derivatives. Modified wave numbers are obtained for various approximations of fractional derivatives. Considering these modified wave numbers, Fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations. A novel fourth order compact scheme is developed for space fractional advection–diffusion reaction equations with variable coefficients in one and two dimensions. In the proposed compact scheme, the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations. This splitting of second derivative approximations allows us to obtain a similar system of linear equations as in Zhao and Tian (2017) which presents only second order accurate finite difference method for advection–diffusion reaction equations in one-dimension. The stability of the proposed compact scheme is demonstrated numerically. Furthermore, the proposed compact scheme is employed to space fractional Black–Scholes equation for pricing European options as an application of fractional derivatives in mathematical finance. Numerical illustrations are presented to validate the theoretical claims.