论文标题
二阶精确数值方案，具有分级网格，用于由粘弹性产生的非线性部分整合差异方程
Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity
论文作者
论文摘要
本文建立并分析了具有弱奇异核的非线性部分集成方程的二阶精确数值方案。在时间方向上，我们将曲柄 - 尼科尔森方法应用于时间导数，并采用产品集成（PI）规则来处理Riemann-Liouville分数积分。从中，使用非均匀的网格来补偿$ t = 0 $的精确解决方案的单数行为，以便我们的方法可以在时间上达到二阶收敛。为了制定完全离散的隐式差异方案，我们为二阶空间衍生物采用标准的中心差公式，而基于分段线性测试功能的Galerkin方法用于近似非线性对流项。然后，我们为提出的隐式差异方案得出数值解的存在和独特性。同时，通过离散能量法证明了稳定性和收敛性。此外，为了证明所提出的方法的有效性，我们利用固定点迭代算法来计算离散方案。最后，数值实验说明了所提出的方案的可行性和效率，其中数值结果与我们的理论分析一致。
This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank-Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann-Liouville fractional integral. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at $t=0$ so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we employ a standard centered difference formula for the second-order spatial derivative, and the Galerkin method based on piecewise linear test functions is used to approximate the nonlinear convection term. Then we derive the existence and uniqueness of numerical solutions for the proposed implicit difference scheme. Meanwhile, stability and convergence are proved by means of the discrete energy method. Furthermore, to demonstrate the effectiveness of the proposed method, we utilize a fixed point iterative algorithm to calculate the discrete scheme. Finally, numerical experiments illustrate the feasibility and efficiency of the proposed scheme, in which numerical results are consistent with our theoretical analysis.
