诺伊曼边界条件下分数阶次扩散方程的紧差分格式

A compact difference scheme for fractional sub-diffusion equations with Neumann boundary conditions

对于诺伊曼边界条件下时间分数阶次扩散方程,提出了紧差分格式,并用该格式数值求解方程。首先,由于该方程在时间为0处解的不光滑性,因此使用非一致网格上的L1格式对时间方向进行离散,一致网格上的紧差分格式对空间方向进行离散,建立紧差分格式;其次,通过离散的能量方法,给出该格式在二范数意义下的收敛性分析;最后,通过Matlab进行数值模拟,验证该格式的有效性。该结果进一步地丰富了分数阶方程的数值算法。

In this paper,a compact difference scheme is proposed for fractional sub-diffusion equations with Neumann boundary conditions,and used to solve the equation numerically.First of all,since the solution of this equation is not smooth when the time is 0,the L1 scheme on the non-uniform mesh is used to discretize the temporal direction,and the compact difference scheme on the uniform mesh is used to discretize the spatial direction,and a compact difference scheme is established.Secondly,a convergence analysis of this scheme in the sense of two-norm is given,with discrete energy method.Finally,numerical simulation is carried out by MATLAB to verify the effectiveness of this scheme.The results further enrich the numerical algorithm of fractional equation.