Optimal error estimate of the Legendre spectral approximation for space-fractional reaction–advection–diffusion equation

Abstract

Abstract In this paper, we consider the space-fractional reaction–advection–diffusion equation with fractional diffusion and integer advection terms. By treating the first-order integer derivative as the composition of two Riemann–Liouville fractional derivative operators, we construct a fully discrete scheme by Legendre spectral method in a spatial and Crank–Nicolson scheme in temporal discretizations. Using thee right Riemann–Liouville fractional derivative, a novel duality argument is established, the optimal error estimate is proved to be O(τ2+N−m) $O(\tau^{2}+N^{-m})$ in L2 $L^{2}$-norm. Numerical tests are carried out to support the theoretical results, and the coefficient matrix with respect to first-order derivative obtained here is compared with that of traditional Legendre spectral method.

Keywords

• Space-fractional reaction–advection–diffusion equation
• Legendre spectral method
• Crank-Nicolson scheme
• Stability
• Optimal error estimate