A boundary integral equation method (BIEM) for dynamic rupture simulation (e.g., Das and Aki 1977) has been widely used in the study of a variety of problems related to earthquake generation processes, from idealized 2-D cases (e.g., Andrews 1985) to 3-D realistic cases dealing with complex geometry composed of many fault planes (e.g., Ando and Kaneko 2018). In addition, the BIEM is a useful tool that can be used to study quasi-static problems (e.g., Okada 1985), the preparation processes of earthquakes (e.g., Dieterich 1992), and their sequences (e.g., Lapusta et al. 2000). It can also be extended to linear viscoelastic materials (Geubelle 1997; Kato 2002; Miyake and Noda 2019).

The BIEM makes use of analytic expressions of Green’s function and does not suffer from artificial dispersion during propagation of elastic waves in a medium. Because we do not have to discretize the medium with the BIEM, it is possible to simulate a problem with higher on-fault resolution than with other methods such as finite difference methods and finite element methods, which are based on discretization of volume. Numerical techniques have been developed for or applied to BIEMs that dramatically reduce computational costs, such as fast Fourier transformation for spectral formulation (Lapusta et al. 2000), fast multipole decomposition (Tullis et al. 2000), a Barnes–Hut scheme (Beeler and Tullis 2011), a fast domain partitioning (FDP) (Ando et al. 2007), hierarchical matrices (H-matrices) (Ohtani et al. 2011), and a recently proposed method consisting of FDP, H-matrices, quantization, and averaged reduced time (Sato and Ando 2019). One of the weak points of BIEMs is that it is difficult to deal with a heterogeneous medium using them, but there are some approaches that extend the ability of BIEMs to such problems (Goto et al. 2010; Kame and Kusakabe 2012). Owing to these efforts, BIEMs are currently one of the most efficient and useful methods for studying fault dynamics in a range of problem settings.

There is, however, an issue with numerical stability in BIEMs (e.g., Fukuyama and Madariaga 1998; Tada and Madariaga 2001). For example, Tada and Madariaga (2001) systematically studied a numerical scheme based on spatiotemporally piecewise-constant basis functions for the rate of displacement gap \( V \) on the fault (Cochard and Madariaga 1994). They simulated a 2-D self-similar dynamic rupture of Mode I, II, or III, and concluded that the region of stability in the numerical parameter space is rather restricted and depends on the mode of the rupture (Figs. 5, 6 and 7 in Tada and Madariaga 2001). The intersection of the stability regimes for multiple modes is so restricted that the numerical parameters must be chosen carefully for simulation of mixed-mode ruptures. Furthermore, the ability of flexible meshing on the surfaces is very restricted, as the numerical parameters vary locally for a heterogeneous mesh. It is often the case that a small artificial damping factor is added (e.g., Yamashita and Fukuyama 1996; Kame et al. 2003) to enhance the stability of the simulations. The effect of the damping should be small, but could make solutions of insufficient resolution artificially smooth and good-looking if simulations were to be tuned unfairly. Therefore, researchers must be careful to avoid misinterpretation of numerical results.

Noda and Lapusta (2010) suggested a different, second-order time-marching scheme based on a predictor–corrector approach for a spectral, wavenumber-domain BIEM (SBIEM) by slightly modifying a scheme proposed by Lapusta et al. (2000) and Lapusta and Liu (2009). In the SBIEM, temporal convolution is calculated with a midpoint integration scheme, and \( V \) at temporal midpoints is estimated by linear interpolation between collocation points. In the present study, we have adapted the midpoint evaluation of \( V \) to the space-domain BIEM based on spatiotemporal piecewise-constant basis functions and found that the present scheme was more stable and accurate than the conventional one.

The outline of this paper is as follows. The two time-marching schemes investigated are briefly described in the “Time-marching schemes” section. The procedures of numerical experiments conducted to study the stability and accuracy of the schemes are described in the “Methodology” section, and the results of these experiments are reported in the “Results” section, which is followed by “Discussion” and “Summary” sections.

### Time-marching schemes

In the present study, we investigated time-marching schemes in the simulation of dynamic rupture propagation for a fault embedded in an infinite homogeneous linear elastic medium. For simplicity we considered 2-D problems with a planer fault subjected to either Mode I, II, or III loading conditions. In such problems, a relevant component of traction on a fault \( \tau \) is expressed as spatiotemporal convolution of an integration kernel \( K \) and history of the rate of displacement discontinuity \( V \) in a boundary integral formulation (Cochard and Madariaga 1994; Geubelle and Rice 1995),

$$ \begin{array}{*{20}c} {\tau \left( {x,t} \right) = \tau^{\text{ini}} + \mathop \smallint \limits_{ - \infty }^{\infty } \mathop \smallint \limits_{0}^{{t_{ - } }} K\left( {x - x^{\prime},t - t^{\prime}} \right)V\left( {x^{\prime},t^{\prime}} \right)dt^{\prime}dx^{\prime} - \eta V\left( {x,t} \right),} \\ \end{array} $$

(1)

where \( x \) is the coordinate along the fault, \( t \) is time, \( \tau^{\text{ini}} \) is the initial value of the traction on the fault if \( V = 0 \). \( t_{ - } \) is infinitesimally smaller than \( t \), and the difference of \( t \) and \( t_{ - } \) is crucial in the proposed scheme. \( K \) is a hypersingular kernel or can be expressed as a combination of non-hypersingular kernels and derivative operators (Cochard and Madariaga 1994; Tada and Yamashita 1997). \( \eta \) represents the amount of an instantaneous elastodynamic effect due to impedance of the medium extracted explicitly from the convolution. It depends on the mode of the rupture,

$$ \begin{array}{*{20}c} {2\eta = \left\{ {\begin{array}{*{20}c} {\rho c_{p} } & {\left( {\text{Mode I}} \right)} \\ {\rho c_{s} } & {\left( {\text{Mode II and III}} \right)} \\ \end{array} } \right. ,} \\ \end{array} $$

(2)

where \( \rho \) is the density, \( c_{s} \) is the S-wave speed, and \( c_{p} \) is the P-wave speed.

### Conventional time-marching scheme

Cochard and Madariaga (1994) discretized Eq. (1) based on piecewise-constant spatiotemporal distribution of \( V \). The collocation points where \( \tau \) and \( V \) are defined are indicated in Fig. 1a. The traction after the \( n \)th timestep and at the \( i \)th element is estimated by the summation of the effects of the sources at the \( m \)th timestep and the \( j \)th element,

$$ \begin{array}{*{20}c} {\tau_{i}^{n} = \tau_{i}^{\text{ini}} + \mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{m = 0}^{n} K_{i - j}^{n - m} \left( {\Delta x,h_{T} ,e_{t} } \right)V_{j}^{m} .} \\ \end{array} $$

(3)

\( n = 0 \) corresponds to the first snapshot defined at \( t = e_{t} \Delta t \). \( N \) is the total number of the elements, \( \Delta x \) is the length of the elements, \( h_{T} \) is the nondimensional timestep \( h_{T} = c_{s} \Delta t/\Delta x \), and \( e_{t} \) is the nondimensional timing of the collocation points (Fig. 1a). \( V_{i}^{n} \) is defined at \( t = \left( {n + e_{t} } \right)\Delta t \) so that the collocation point is in the interior of the support of the corresponding basis function if \( 0 < e_{t} < 1 \). Note that the kernel value depends only on the relative location and time difference between a source and a receiver in our simple problem setting. The instantaneous term in Eq. (1) is included in the summation by assuming that \( V \) does not change suddenly at the collocation points. For expression of the kernel, please see Appendix in Tada and Madariaga (2001).

Suppose that the values of \( V_{j}^{m} \) are known for all \( j \) and \( m \le n - 1 \). Then Eq. (3) yields \( N \) equations with \( 2N \) unknowns (\( \tau_{i}^{n} \) and \( V_{i}^{n} \), \( 1 \le i \le N \)). In problems dealing with rupture propagation, a fault constitutive law is assumed. As it must satisfy the principle of local action, it provides additional \( N \) equations that relate \( \tau_{i}^{n} \) and \( V_{i}^{n} \). Therefore, we can solve for \( \tau_{i}^{n} \) and \( V_{i}^{n} \), if they exist, and carry on the simulation. This scheme shall be abbreviated as CM in the present paper.

CM has two nondimensional numerical parameters \( h_{T} \) and \( e_{t} \), and Tada and Madariaga (2001) demonstrated that the numerical stability depends on the selection of them, and on the mode of the rupture. Their results are compared with our numerical experiments in a later section.

### Present scheme based on a predictor–corrector method

Noda and Lapusta (2010) proposed a second-order time-marching scheme based on a temporal midpoint integration scheme of Eq. (1) formulated in a wavenumber domain for use with the SBIEM (e.g., Lapusta et al. 2000). In the SBIEM, Fourier basis is used so that the spatial distribution of \( V \) is assumed to be smooth. Such representation is available only for a planer fault, which is recognized as a weakness of the SBIEM. Here, we applied a time-marching scheme similar to that by Noda and Lapusta (2010) to the space-domain BIEM.

The spatiotemporal discretization in the present schema is illustrated in Fig. 1b. We approximate Eq. (1) using spatiotemporally piecewise-constant basis functions for \( V \) similarly to CM, but estimate the history of \( V \) at temporal midpoints indicated by crosses, and allow for sudden change in \( V \) at the collocation points indicated by circles.

$$ \begin{array}{*{20}c} {\tau_{i}^{n} = \tau_{i}^{\text{ini}} + \mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{m = 0}^{n - 1} K_{i - j}^{n - m} \left( {\Delta x,h_{T} ,1 + } \right)V_{j}^{m + 1/2} - \eta V_{i}^{n} .} \\ \end{array} $$

(4)

\( V_{i}^{n} \) and \( V_{i}^{m + 1/2} \) are defined at time \( n\Delta t \) and \( \left( {m + 1/2} \right)\Delta t \), respectively. It should be noted that we use the same integration kernel as in CM. \( 1 + \) in Eq. (4) is infinitesimally larger than \( 1 \), and explicitly indicate that the collocation point for \( \tau_{i}^{n} \) and \( V_{i}^{n} \) is out of the support of the basis function for \( V_{i}^{n - 1/2} \). Because the timing of the collocation point relative to the basis function is fixed, this discretization has only one nondimensional parameter \( h_{T} \).

After the \( n - 1 \)th timestep, we know the \( V_{i}^{m} \) values where \( m \le n - 1 \) from which we can calculate the contribution from the previous history to the functional term,

$$ \begin{array}{*{20}c} {\phi_{i}^{\text{tail}} = \tau_{i}^{\text{ini}} + \mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{m = 0}^{n - 2} K_{i - j}^{n - m} \left( {\Delta x,h_{T} ,1 + } \right)V_{j}^{m + 1/2} .} \\ \end{array} $$

(5)

The traction at the end of the current timestep is expressed as

$$ \begin{array}{*{20}c} {\tau_{i}^{n} = \phi_{i}^{\text{tail}} + \phi_{i}^{\text{head}} - \eta V_{i}^{n} ,} \\ \end{array} $$

(6)

where

$$ \begin{array}{*{20}c} {\phi_{i}^{\text{head}} = \mathop \sum \limits_{j = 1}^{N} K_{i - j}^{1} \left( {\Delta x,h_{T} ,1 + } \right)V_{j}^{n - 1/2} .} \\ \end{array} $$

(7)

In a predictor step, we assume \( V_{j}^{n - 1/2} \) to be identical to \( V_{j}^{n - 1} \), and estimate \( \phi_{i}^{\text{head}} \),

$$ \begin{array}{*{20}c} {V_{*j}^{n - 1/2} = V_{j}^{n - 1} ,} \\ \end{array} $$

(8)

$$ \begin{array}{*{20}c} {\phi_{*i}^{\text{head}} = \mathop \sum \limits_{j = 1}^{N} K_{i - j}^{1} \left( {\Delta x,h_{T} ,1 + } \right)V_{*j}^{n - 1/2} .} \\ \end{array} $$

(9)

This is a first-order approximation if the solution is smooth. We can then obtain an approximation to \( \tau_{i}^{n} \) and \( V_{i}^{n} \) by solving a friction law and

$$ \begin{array}{*{20}c} {\tau_{*i}^{n} = \phi_{i}^{\text{tail}} + \phi_{*i}^{\text{head}} - \eta V_{*i}^{n} } \\ \end{array} $$

(10)

simultaneously. If we adopted \( \tau_{*i}^{n} \) and \( V_{*i}^{n} \) as the final estimate and \( V_{j}^{n - 1/2} \) were not updated, the scheme would be CM with \( e_{t} = 0 + \).

In a corrector step, we first obtain a better estimate of \( V_{j}^{n - 1/2} \) based on linear interpolation,

$$ \begin{array}{*{20}c} {V_{**j}^{n - 1/2} = \frac{1}{2}\left( {V_{j}^{n - 1} + V_{*j}^{n} } \right),} \\ \end{array} $$

(11)

and re-calculate the contribution of the current timestep to the functional term,

$$ \begin{array}{*{20}c} {\phi_{**i}^{\text{head}} = \mathop \sum \limits_{j = 1}^{N} K_{i - j}^{1} \left( {\Delta x,h_{T} ,1 + } \right)V_{**j}^{n - 1/2} .} \\ \end{array} $$

(12)

Then, we update the estimate of \( \tau_{i}^{n} \) and \( V_{i}^{n} \) by solving the friction law and

$$ \begin{array}{*{20}c} {\tau_{**i}^{n} = \phi_{i}^{\text{tail}} + \phi_{**i}^{\text{head}} - \eta V_{**i}^{n} .} \\ \end{array} $$

(13)

Finally, we adopt these updated values as a numerical solution,

$$ \begin{array}{*{20}c} {V_{i}^{n} = V_{**i}^{n} ,} \\ \end{array} $$

(14)

$$ \begin{array}{*{20}c} {\tau_{i}^{n} = \tau_{**i}^{n} ,} \\ \end{array} $$

(15)

and

$$ \begin{array}{*{20}c} {V_{i}^{n - 1/2} = \frac{1}{2}\left( {V_{i}^{n - 1} + V_{**i}^{n} } \right) .} \\ \end{array} $$

(16)

This scheme shall be labeled as NL in this paper.

If the corrector step is iterated until the numerical solution converges, the scheme becomes implicit, which solves Eq. (4) and the friction law simultaneously. For a relatively short timestep (\( {{h_{T} c_{s} } \mathord{\left/ {\vphantom {{h_{T} c_{s} } {c_{p} }}} \right. \kern-0pt} {c_{p} }} \le 0.5 \) for Modes I and II, and \( h_{T} \le 0.5 \) for Mode III), \( \phi_{i}^{\text{head}} \) is zero regardless of \( V_{j}^{n - 1/2} \). Therefore, the numerical solution converges after the first corrector step.

The computational cost is dominated by calculation of spatiotemporal convolution. Although this scheme consists of two sub-steps, the main part of the convolution, summing up contributions from the previous steps, is executed only once per timestep. Therefore, the additional numerical resources required are negligible relative to CM.

\( V \) is assumed to be constant in the predictor step. We can interpret the corrector step as two-step approximation. Firstly, \( V \) is assumed to change linearly with time, which would yield global error scaled as \( \Delta t^{2} \). Secondly, the mid-point evaluation of \( V \) (Eqs. 11 and 16) is applied and the same kernel as for CM is used instead of preparing another integration kernel for the linearly evolving \( V \). This second approximation also yields global error scaled as \( \Delta t^{2} \) where \( K \) is smooth enough (Appendix). Just note that in the SBIEM by Noda and Lapusta (2010), both \( V \) and the integration kernel are evaluated at the temporal midpoints so that the temporal integration is estimated with the midpoint scheme. In the numerical experiments presented later, NL was shown to be useful owing to its enhanced numerical stability, faster convergence of numerical error, and minimal modification of the simulation code required relative to CM.