A compact finite difference method for solving a class of time fractional convection-subdiffusion equations

Abstract

A high-order compact finite difference method is proposed for solving a class of time fractional convection-subdiffusion equations. The convection coefficient in the equation may be spatially variable, and the time fractional derivative is in the Caputo’s sense with the order \(\alpha \) (\(0<\alpha <1\)). After a transformation of the original equation, the spatial derivative is discretized by a fourth-order compact finite difference method and the time fractional derivative is approximated by a \((2-\alpha )\)-order implicit scheme. The local truncation error and the solvability of the method are discussed in detail. A rigorous theoretical analysis of the stability and convergence is carried out using the discrete energy method, and the optimal error estimates in the discrete \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained. Applications using several model problems give numerical results that demonstrate the effectiveness and the accuracy of this new method.