分数阶扩散方程的一种新的高阶数值方法(英文)

A New High Order Numerical Method for the Fractional Diffusion Equation

研究了分数阶扩散方程具有初边值问题的数值解法.基于Riemann-Liouville分数阶导数的定义,直接对该方程采取积分离散,利用四阶紧致有限差分算子对空间二阶导数近似,得到此方程的高阶隐式格式.证明了该格式是唯一可解的,并采用Fourier方法证明了该隐式格式是无条件稳定的.进一步,利用线性插值的方法提高了格式的误差阶,从所给的数值结果可以看出,改进后的格式的误差阶可达到O(τ2+h4).

In this paper,we consider numerical solving for a fractional diffusion equation（FDE） with initial condition and Dirichlet boundary condition.A higher-order implicit scheme for the FDE is presented.By means of Fourier method,we provide the stability analysis of the implicit scheme.The local truncation error for the scheme is O（τ1＋γ＋τγh4）.Moreover,we improve the temporal accuracy for the implicit finite difference scheme.The local truncation error of theimproved scheme can reach O（τ2＋h4）.At the end of the paper,we give a numerical example to demonstrate the theoretic analysis.