论文标题
$ \ mathbb {r}^2中的快速和高阶精度数值方法
Fast and High-order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in $\mathbb{R}^2
论文作者
论文摘要
在本文中，我们研究了曲柄 - 尼古尔森方法，用于时间维度和分段二次多项式搭配方法，用于时间依赖时间依赖的非本地问题的空间维度。这种离散化的新理论结果是，所提出的数值方法是无条件稳定的，其全局截断误差为$ \ Mathcal {o} \ left（τ^2+h^{4-γ} \ right）$，$ 0 <γ<1 $，其中$τ$和$ h $是iNVISIAL SPAT and Spatials and Spatials and Spatials and Spatials and Spatials and spat。另外，我们开发了共轭梯度平方方法来解决由时间依赖性的非本地问题（包括二维病例）引起的产生的离散的非对称和无限期系统。通过在非本地问题中使用添加剂和乘法库奇内核，可以在共轭梯度平方迭代中有效地执行结构化系数矩阵矢量乘法。给出了数值示例来说明我们的理论结果，并证明所提出的方法的计算成本为$ O（M \ log M）$操作，其中$ m $是套在一起点的数量。
In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $\mathcal{O}\left(τ^2+h^{4-γ}\right)$ with $0<γ<1$, where $τ$ and $h$ are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in non-local problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of $O(M \log M)$ operations where $M$ is the number of collocation points.
