非齐次边界空间分数阶扩散方程的数值方法

Numerical Methods for Fractional Diffusion Equations in Non-Homogeneous Boundary Spaces

摘要: 针对带非齐次Dirichlet边界条件的一维变系数非线性双侧空间分数阶扩散方程，本文运用并分析了一种高效稳定的数值求解方法。首先，通过引入适当的变量变换，将原问题转化为齐次边界形式；在此基础上，时间方向采用具有二阶精度的半隐式中心差分格式，空间方向利用加权移位公式对双侧分数阶导数进行离散，从而建立了一种全离散有限差分格式。随后，在满足非线性项局部Lipschitz条件的假设下，严格证明了该数值格式的无条件稳定性和二阶收敛性。在数值实现过程中，由于空间分数阶算子的非局部特性，时间推进的每一时间层均需反复求解非对称稠密线性代数方程组。为提高整体计算效率，本文采用了带Jacobi预处理的Bi-CGSTAB方法对线性系统进行求解，并与传统的Bi-CGSTAB方法进行了对比。数值实验结果表明，在相同精度要求下，该方法显著降低了平均迭代次数，整体求解格式呈现出网格无关收敛性，并在长时间模拟中表现出明显的CPU时间优势。综上，本文采用的预处理Bi-CGSTAB方法在精度、稳定性和计算效率方面均具有良好表现，为一类非线性空间分数阶扩散方程的高效数值求解提供了可行方案。

Abstract: For the 1D variable-coefficient nonlinear bilateral space fractional diffusion equation with non-homogeneous Dirichlet boundary conditions, this paper applies and analyzes an efficient and stable numerical method. Firstly, by introducing an appropriate variable transformation, the original problem is transformed into a homogeneous boundary form. Based on this, a second-order semi-implicit central difference scheme is adopted in time, and a weighted shifted formula is used to discretize the bilateral fractional derivatives in space, thereby establishing a fully discrete finite difference scheme. Subsequently, assuming the nonlinear term satisfies a local Lipschitz condition, the unconditional stability and second-order convergence of the numerical scheme are rigorously proven. In the numerical implementation, due to the non-local nature of the spatial fractional operator, each time step requires repeatedly solving asymmetric dense linear algebraic systems. To improve overall computational efficiency, this paper employs the Jacobi-preconditioned Bi-CGSTAB method to solve the linear system, comparing it with the traditional Bi-CGSTAB method. Numerical results show that, under the same accuracy requirements, this method significantly reduces the average number of iterations. The overall scheme exhibits grid-independent convergence and demonstrates obvious CPU time advantages in long-time simulations. In summary, the adopted preconditioned Bi-CGSTAB method performs well in accuracy, stability, and computational efficiency, providing a feasible approach for the efficient numerical solution of a class of nonlinear spatial fractional diffusion equations.