An efficient second-order convergent scheme for one-side space fractional diffusion equations with variable coefficients
It addresses the bottleneck of low-order convergence in existing schemes for fractional diffusion equations with variable coefficients, offering a more accurate and efficient numerical method.
The paper develops a second-order finite difference scheme for one-side space fractional diffusion equations with variable coefficients, achieving unconditional stability and second-order convergence in both time and space, with a preconditioner that ensures linear convergence of Krylov subspace solvers.
In this paper, a second order finite difference scheme is investigated for time-dependent one-side space fractional diffusion equations with variable coefficients. The existing schemes for the equation with variable coefficients have temporal convergence rate no better than second order and spatial convergence rate no better than first order, theoretically. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the diffusion coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.