The boundary integral equation method (BIEM) is one of the important numerical techniques used to simulate geophysical phenomena including dynamic propagation, nucleation, and sequence of earthquake ruptures. We studied the stability and convergence of two time-marching schemes numerically for 2-D problems in Mode I, II, and III conditions. One was a conventional method based on piecewise-constant spatiotemporal distribution of the rate of displacement gap $$ V $$ V (CM), and the other was a slightly modified scheme from a predictor–corrector method previously applied to a spectral BIEM (NL). In the stability analysis, we simulated behavior of a traction-free fault under uncorrelated random distributions of initial traction. The growth rate of the perturbation is negative in a parameter regime of complex shape with CM, which has two numerical parameters, and the intersection for all the modes is very restricted as reported previously. In contrast, NL has only one parameter and yields simpler and a wide parameter regime of stability, conceivably allowing more flexible meshing on the fault. In the convergence analysis in which a smooth problem was solved, CM resulted in a numerical error scaled as $$ \Delta x^{1} $$ Δ x 1 while NL led to the scaling of $$ \Delta x^{2} $$ Δ x 2 typically or of $$ \Delta x^{1.5} $$ Δ x 1.5 under certain conditions in Mode II problems. NL requires negligible additional computational costs and modification of the code is quite straightforward relative to CM. Therefore, we conclude that NL is a useful time-marching scheme that has wide applicability in simulations of earthquake ruptures although the reason for the rather complicated convergence behavior and verification of the findings here to more general conditions deserve further study.

中文翻译：

---

使用空间域 BIEM 进行动态破裂模拟的两种时间推进方案的比较

边界积分方程法（BIEM）是一种重要的数值技术，用于模拟地球物理现象，包括动态传播、成核和地震破裂序列。我们对模式 I、II 和 III 条件下的二维问题的两个时间推进方案的稳定性和收敛性进行了数值研究。一种是基于位移间隙速率 $$ V $ $ V (CM) 的分段恒定时空分布的传统方法，另一种是对先前应用于光谱 BIEM (NL ）。在稳定性分析中，我们模拟了初始牵引不相关随机分布下无牵引故障的行为。在具有 CM 的复杂形状的参数范围内，扰动的增长率为负，它有两个数值参数，并且所有模式的交集非常受限，如前所述。相比之下，NL 只有一个参数并产生更简单和更广泛的稳定参数范围，可以想象允许更灵活的故障网格划分。在求解平滑问题的收敛性分析中，CM 导致数值误差缩放为 $$ \Delta x^{1} $$ Δ x 1 而 NL 导致 $$ \Delta x^{2} $$ Δ x 2 通常或 $$ \Delta x^{1.5} $$ Δ x 1.5 在模式 II 问题的某些条件下。NL 需要的额外计算成本可以忽略不计，并且相对于 CM，代码的修改非常简单。所以，