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 \({\mathscr {O}}\left( \tau ^2+h^{4-\gamma }\right) \) with \(0<\gamma <1\), where \(\tau \) 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 nonlocal 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.

在本文中，我们研究时间相关的非局部问题的空间维数的Crank-Nicolson方法和空间维的分段二次多项式配置方法。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\（{\ mathscr {O}} \ left（\ tau ^ 2 + h ^ {4- \ gamma} \ right ）\）与\（0 <\ gamma <1 \），其中\（\ tau \）和h是分别在时间和空间维度上的离散化大小。我们还开发了共轭梯度平方方法来解决由离散的非时间性局部问题（包括二维情况）引起的离散化不对称和不定系统。通过在非局部问题中使用加性和乘性柯西核，可以在共轭梯度平方迭代中高效执行结构化系数矩阵-矢量乘法。数值算例说明了我们的理论结果，并证明了所提出方法的计算成本为\（O（M \ log M）\）运算，其中M为配置点数。