基于适当正交分解技术的空间-分数扩散方程的降维加权显式有限差分法

Axioms Pub Date : 2024-07-08 DOI:10.3390/axioms13070461
Xuehui Ren, Hong Li
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摘要

讨论了一种基于加权显式有限差分方案和适当正交分解(POD)技术的空间黎曼-刘维尔分数导数扩散方程的降维方法。所构建的以矩阵形式编写的近似方法不仅能确保足够的精度阶次,还能减少自由度、降低存储要求并加快计算速度。通过矩阵分析证明了唯一性、稳定性和误差估计。概述了 POD 算法的程序步骤,该算法可降低维度。通过数值模拟评估了降维加权显式有限差分法的可行性和有效性。比较了降维方法和经典的加权显式有限差分方案,包括 L2 准则误差、精度阶次和 CPU 时间。
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A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation
A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the L2 norm, the accuracy order, and the CPU time.
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