Fast dissipation-preserving difference scheme for nonlinear generalized wave equations with the integral fractional Laplacian
Fast dissipation-preserving difference scheme for nonlinear generalized wave equations with the integral fractional Laplacian
复制标题
积分分数拉普拉斯非线性广义波动方程的快速保耗散差分格式
DOI：
10.1016/j.cnsns.2021.105786
复制
发表时间：
2021
影响因子：
3.9
通讯作者：
Wang Yushun
中科院分区：
文献类型：
--
作者：
Hu Dongdong;Cai Wenjun;Fu Yayun;Wang Yushun
本文构造了二维非线性广义波动方程的积分分数阶差分格式。详细讨论了该方案的离散耗散性质，并分析了该方案的存在唯一性和无条件收敛性。进一步，我们发现空间离散产生一个块toeplitz系数矩阵，并且随着空间网格网格数M和分数阶α的增加，它将是病态的。因此，我们对非线性系统开发了一种有效的线性化迭代算法，使得它可以通过具有适当前置条件的Krylov子空间求解器有效地求解，其中求解器中使用二维快速傅里叶变换来加速矩阵向量乘积。大量的数值实验验证了该方案的理论分析和长期计算的离散耗散规律。
In this paper, we construct a dissipation-preserving difference scheme for two-dimensional nonlinear generalized wave equations with the integral fractional Laplacian. We discuss the discrete dissipation property of the scheme in detail, and we also analyze the existence, uniqueness and the unconditional convergence of the proposed scheme. Further, we reveal that the spatial discretization generates a block-Toeplitz coefficient matrix, and it will be ill-conditioned as the spatial grid mesh number M and the fractional order α increase. Thus, we exploit an efficient linearized iteration algorithm for the nonlinear system, such that it can be efficiently solved by the Krylov subspace solver with a suitable preconditioner, where the two-dimensional fast Fourier transform is used in the solver to accelerate the matrix-vector product. Extensive numerical experiments are provided to verify the theoretical analysis and discrete dissipation law of the proposed scheme in long-time computations.