A compact finite difference method for a class of time fractional convection-diffusion-wave equations with variable coefficients

This paper is concerned with numerical methods for a class of time fractional convection-diffusion-wave equations. The convection coefficient in the equation may be spatially variable and the time fractional derivative is in the Caputo sense with the order α (1 < α < 2). The class of the equations includes time fractional convection-diffusion-wave/diffusion-wave equations with or without damping as its special cases. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference method for the spatial derivative and by the L₁ approximation coupled with the Crank-Nicolson technique for the time derivative. 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 a discrete energy analysis method. The optimal error estimates in the discrete H¹, L² and L^∞ norms are obtained under the mild condition that the time step is smaller than a positive constant, which depends solely upon physical parameters involved (this condition is no longer required for the special case of constant coefficients). Applications using three model problems give numerical results that demonstrate the effectiveness and the accuracy of the proposed method.