论文标题
二维非线性Volterra Integro-Differentix方程，具有弱奇异内核
A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel
论文作者
论文摘要
在本文中，提出了一种具有弱奇异内核的二维非线性伏特拉差异方程的两芯时间二阶方案，以减少计算时间并提高Xu等人开发的方案的准确性。 （应用数学数学152（2020）169-184）。所提出的方案包括三个步骤：首先，使用固定点迭代在粗网格上求解一个小的非线性系统。其次，Lagrange的线性插值公式用于得出一些辅助值以分析细网格。最后，在细网格上求解了线性化的曲柄 - 尼科尔森有限差系统。此外，该算法使用空间衍生物的中心差近似。在时间方向上，分别通过曲柄 - 尼古尔森技术和产品积分规则近似时间衍生词和积分项。借助离散能量法，在$ l^2 $ norm中获得了拟议方法的稳定性和时空二阶收敛。最后，数值结果与理论分析一致，并验证算法的有效性。
In this paper, a two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel is proposed to reduce the computation time and improve the accuracy of the scheme developed by Xu et al. (Applied Numerical Mathematics 152 (2020) 169-184). The proposed scheme consists of three steps: First, a small nonlinear system is solved on the coarse grid using fix-point iteration. Second, the Lagrange's linear interpolation formula is used to arrive at some auxiliary values for analysis of the fine grid. Finally, a linearized Crank-Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by Crank-Nicolson technique and product integral rule, respectively. With the help of the discrete energy method, the stability and space-time second-order convergence of the proposed approach are obtained in $L^2$-norm. Finally, the numerical results agree with the theoretical analysis and verify the effectiveness of the algorithm.
