An efficient implicit-explicit adaptive time stepping scheme for multiple-time scale problems in shear zone development


  • The copyright line for this article was changed on 6 May 2015 after original online publication.


[1] Problems associated with shear zone development in the lithosphere involve features of widely different time scales, since the gradual buildup of stress leads to rapid and localized shear instability. These phenomena have a large stiffness in time domain and cannot be solved efficiently by a single time-integration scheme. This conundrum has forced us to use an adaptive time-stepping scheme, in particular, the adaptive time-stepping scheme (ATS) where the former is adopted for stages of quasi-static deformation and the latter for stages involving short time scale nonlinear feedback. To test the efficiency of this adaptive scheme, we compared it with implicit and explicit schemes for two different cases involving: (1) shear localization around the predefined notched zone and (2) asymmetric shear instability from a sharp elastic heterogeneity. The ATS resulted in a stronger localization of shear zone than the other two schemes. We report that usual implicit time step strategy cannot properly simulate the shear heating due to a large discrepancy between rates of overall deformation and instability propagation around the shear zone. Our comparative study shows that, while the overall patterns of the ATS are similar to those of a single time-stepping method, a finer temperature profile with greater magnitude can be obtained with the ATS. The ability to model an accurate temperature distribution around the shear zone may have important implications for more precise timing of shear rupturing.

1. Introduction

[2] Lithospheric rupture is essential to plate tectonics, and yet many aspects of this crucial event are not well understood [e.g., Scholz, 2002]. In particular, little is known about the initiation of shear zone, which acts as a weak zone within lithosphere near many large deformation zones. The understanding of the development of shear zone is essential to elucidate the cause of subduction initiation [Regenauer-Lieb et al., 2001] and slab detachments [Gerya et al., 2004]. One of the favorite explanations for the development is that they begin by concentration of stress at a localized region and then grow by positive feedback between shear heating and reduction in material strength [e.g., Bercovici, 2002; Branlund et al., 2000; Hobbs et al., 2007]. However, the effective simulation of these features has always been a challenge, because of the multiple-time and spatial scales involved in this extremely nonlinear problem.

[3] The process that leads to lithospheric shear zone can in general be divided into three stages (see the cartoons in Figure 1). The stage 1 involves the buildup of stress by tectonic loading. In stage 2, plastic deformation starts to occur and then thermal instability develops within a narrow zone in the lithosphere. The stage 3 can be envisaged as a period where the temperature in the localized zone becomes stable as the heat generated at the zone is balanced by thermal diffusion [Kincaid and Silver, 1996].

Figure 1.

The cartoons showing three stages in shear zone development. Red color means high temperature or plastic strain rate. Gray color represents small variation in temperature of strain rate. Stages 1 and 3 are dominated by relatively long timescale physics, whereas stage 2 is under the short timescale nonlinear physics of the coupling between momentum, constitutive, and energy equations.

[4] A particular difficulty when simulating the development of shear zone is that it contains features with vastly different time scales and spatial scales. For instance, the entire domain where the force is being applied is substantially large, whereas the area of significant deformation can be quite localized. Also during most of computing time of the numerical experiment, the whole region may deform steadily in the stages 1 and 3 mentioned above, which contrasts with the abrupt development of shear instability in the stage 2. Since there is a large difference in characteristic time scales for each stage, the numerical formulation is not easy and falls under the category of large stiffness problems [Dahlquist and Björck, 2008].

[5] To resolve the large discrepancies in spatial scales, in recent years, more and more problems employ adaptive mesh refinement to calculate growing or instabilities with moving boundaries [e.g., Stadler et al., 2010]. However, for problems with large differences in time scales, there appears to be no simple solution. Up to now, most studies adopt a single time-stepping approach (i.e., implicit or explicit schemes). Employing a single scheme may be convenient, but as we shall demonstrate, it may miss short time scale features, which can be important for understanding the intricate physics around the localized zone.

[6] As an attempt to handle the dilemma with different time scales, we propose the use of the implicit-explicit time-integration method [Brown, 2011; Constantinescu and Sandu, 2010]. This method can be divided into two different kinds of schemes. One is implicit-explicit combined scheme, where the implicit and explicit schemes are respectively used for advection and diffusion terms [Constantinescu and Sandu, 2010]. The second case is the adaptive scheme, which switches between implicit and explicit schemes when dominant time scales and mathematical stiffness are changed abruptly with time [Butcher, 1990; Hairer and Wanner, 2004]. We have focused on the adaptive time-stepping scheme (ATS) where the former scheme is applied to the slow deforming phase and the latter to fast propagation of the instability. By doing so, we exploit the advantages of each scheme.

[7] ATS has the potential to calculate accurately multiple time scale phenomena. However, this approach has not been widely adopted in geodynamical simulations. In this study, we compare the ATS against the two previous investigations of the shear zone development [Regenauer-Lieb and Yuen, 1998; So et al., 2012] to see whether the ATS provides a better solution than previous approach using a single time-stepping method. In the case of Regenauer-Lieb and Yuen [1998] (hereinafter referred to as R model), instabilities are triggered around the predefined notched hole as a result of far-field extension, whereas So et al. [2012] (hereinafter referred to as S model) examined the development of asymmetric instability generated at the interface between the stiff and soft lithospheres by far-field compression. In addition, we conducted two benchmark tests to ensure that solutions obtained from our numerical techniques are consistent with a simple analytical solution (benchmark test I) and Ogawa's model [Ogawa, 1987].

[8] Our study shows that for modeling multiscale problems the ATS approach is better than one based on a single time-stepping method in terms of its accuracy and ability to handle highly nonlinear thermal-mechanical feedback. In the two examples [Regenauer-Lieb and Yuen, 1998; So et al., 2012] that were considered, the results of the ATS show fine-scale features near the localized shear zone that were difficult to be observed using the implicit scheme alone.

2. General Model Setup

[9] We used ABAQUS [Hibbit, Karlsson and Sorenson Inc., 2009] a finite element code, which allows the user to prescribe either implicit or explicit time-stepping method. The solvers can be set to have the same order of accuracy for both approaches. The use of this particular code was necessary because the two previous studies (R and S models) employed the implicit scheme.

[10] We assumed that the mass, momentum, and energy are conserved within the system which is made up of a material whose strength is stress- and temperature-dependent [Glen, 1955; Karato, 2008]. Equations (1)-(3) represent the continuity equation, objective Jaumann derivative of the stress tensor [Kaus and Podladchikov, 2006], and energy equation, respectively. D/Dt is the material derivative.

display math(1)
display math(2)
display math(3)
display math(4)

where t is the time, xi is the spatial coordinate along the i direction and vi is the velocity in i direction. τij and Wij are deviatoric stress and spin rate tensors as defined by equation (4), respectively. The detailed meaning and value of the parameters for different cases are listed in Table 1. Other information concerning the model, such as mesh size and initial condition, can be found in Table 2.

Table 1. Input Parameters of Assessments for Three Models
VariablesSymbol (Unit)Ogawa's ModelR ModelS Model
Specific heatcp (J/(kg·K))80012401240
Thermal conductivityk (W/(m·K))
Densityρ (kg/m3)300033003300
Shear modulusμ (Pa)Use Young's modulusUse Young's modulusVariable
Young's modulusK (Pa)7 × 10101011Use shear modulus
Activation energyQ (kJ/mol)500498498
Universal gas constantR (J/(K·mol))8.3148.3148.314
Power law exponentn34.484.48
PrefactorA (Pa−n·s−1)4.3 × 10−165.5 × 10−255.5 × 10−25
Yield strengthσyield (MPa)Already yielded100100
Poisson's ratiov0.30.30.3
The convergence efficiency from plastic work into shear heatingΨ10.90.9
Table 2. Differences of the Three Models Being Assessed
VariablesOgawa's ModelR ModelS Model
Material typeHomogeneous viscoelasticHomogeneous elastoplasticBimaterial elastoplastic
Size of elements0.05 kmFine part: 0.25 km × 0.25 kmFine part: 0.2 km × 0.25 km
  Coarse part: 1 km × 0.25 kmcoarse part: 1 km × 0.25 km
Number of grid points∼2000∼450,000∼600,000
RheologyStrain rate and stress dependentSameSame
Predefined weak zoneNoNotched holeNo
Initial temperature fieldUniformly 978 KUniformly 978 KUniformly 978 K
Nondimensionalization?yes [see Ogawa, 1987]NoNo
Boundary velocity (cm/yr)0.3–9.0 cm/yr2.0–8.0 cm/yr2.0 cm/yr
Yield criterionNovon Misesvon Mises

[11] We also assumed that

display math(5)
display math(6)


display math(7)

where inline image (equation (5)) is total strain-rate tensor defined by the simple sum of elastic and inelastic strain-rate tensors (equation (6)). The former can be expressed as in equation (7), and μ denotes the elastic modulus such as the Young's (in the cases of Ogawa's and R model) or shear modulus (in the case of S model).

display math(8)
display math(9)

[12] When the second invariant of deviatoric stress tensor J2 (equation (8)) of each node exceeds the predefined yield strength (σyield), the lithosphere is assumed to behave in an inelastic manner as a function of deviatoric stress and temperature (equation (9)). Moreover, this inelastic strain is converted into shear heating with a ratio of Ψ (see equations (3) and (9)). In this study, we use von Mises yield criterion with 100 MPa of finite yield strength. Both plastic deformation and frictional motion cause temperature elevation in the lithosphere. Rheologies for both mechanisms are different, but large plastic yield strength (∼100 MPa) has been used to mimic a strong fault [Hale et al., 2010], which has a large frictional coefficient (f ≈ 0.7) and causes a great deal of heat generation.

3. Description of Different Numerical Schemes

3.1. Explicit Scheme

[13] The general mathematical formulation for the explicit scheme can be simply described as

display math(10)

[14] In this scheme, the information on the next time step inline image comes directly from inline image and the function f which comes from the discretization of the partial differential equation for the system at hand [Griffiths and Higham, 2010]. It is quite straightforward. However, the explicit scheme can become unstable because of the relationship between time step and spatial mesh, and thus to avoid such ill behavior the size of time step should obey a strict criterion according to numerical analysis [e.g., Dahlquist and Björck, 2008]. This restriction poses a severe problem in geodynamical modeling where the solution has to be integrated over a very long period of time in which case the implicit scheme must be used. The detailed formulation for the discretization and the criterion for time step are presented in section 'Explicit Scheme'.

3.2. Implicit Scheme

[15] The implicit time stepping scheme, on the other hand, can be expressed as below

display math(11)

where inline image is the unknown (i.e., temperature or displacement at each node) at time t + Δt at ith iteration. f is the discretized functional derived from partial differential equations. Unlike the explicit scheme, inline image is calculated from both inline image and inline image. inline image is generally obtained using an iterative method which is continued until the difference between inline image and inline image becomes small enough to ensure local convergence. As mentioned previously, the main advantage of the implicit scheme is that rather large time steps can be taken without worrying about the solution becoming unstable. However, the disadvantage is that it can often miss short time scale features and is not suitable for handling highly dynamic circumstances. The implicit scheme is the method of choice for steady state situations [King et al., 2010]. We provide a detailed description of the implicit scheme in section 'Implicit Scheme'.

3.3. ATS Approach

[16] Many geodynamical problems involve features with vast time scale differences and thus may not be suitable for solving them using either explicit or implicit approach. In the previous works on the shear localization in crystalline structure [e.g., Braeck and Podladchikov, 2007] and bimaterial interface [e.g., Langer et al., 2010], time steps and schemes were varied to increase the accuracy of the calculations of highly nonlinear physics. However, there is no detailed investigation for an algorithm to determine the time step and the time-integration scheme.

[17] For these sets of problems, the adaptive time stepping scheme (simply referred to as the ATS) can be a solution to these tough situations [e.g., Butcher, 1990; Hairer and Wanner, 2004]. For instance, as mentioned above, the process of lithospheric shear zone development can be divided into different stages depending on the dominant physics (see Figure 1). The implicit scheme may be suitable for the stages 1 and 3 where the rate of deformation and change in temperature are relatively small and steady. On the other hand, for the stage 2 where the change in temperature and material strength is relatively abrupt, the explicit scheme is a better approach.

[18] In geodynamical problems dealing with shear heating, the velocity of shear instability propagation is much faster than the deformation rate [e.g., So et al., 2012]. The deformation rate is relatively steady while the thermal instability is suddenly initiated and propagated. The lithospheric system is significantly influenced by the thermal event (i.e., shear heating) arisen from energy equation. Moreover, if we adopted the explicit scheme for momentum equation, the size of time step is less than a second. This extremely small time step would be too costly. Therefore, we switch between the implicit and explicit schemes only for energy equation, while keeping the implicit scheme for momentum equation.

4. Benchmark Tests

[19] We should ideally compare the numerical results with analytical solution. However, in our problem where the momentum and energy equations are coupled through stress- and temperature-dependent nonlinear rheology, analytical solution to our best knowledge is not available. In order to demonstrate the strength and weakness of individual schemes and reliability of our techniques in handling this type of challenging problem, two benchmark tests were made. Benchmark test I is a case where the numerical results of the implicit and explicit schemes are compared with the known analytical solution involving shear heating within purely viscous lithosphere with a temperature-dependent viscosity. Finally, we carry out benchmark test for Ogawa's model with numerical results from Ogawa [1987], which employed the explicit method.

4.1. Benchmark Test I

[20] The steady state analytical solution for a viscous fluid with the temperature-dependent viscosity was derived by Sukanek et al. [1973]. Turcotte and Schubert [2002] extended the problem to include large geological spatial scales. We compare the numerically generated solutions from the explicit and implicit schemes with the analytical solution. Our results show clearly that the explicit scheme is more appropriate for dealing with the shear deformation alongside strong feedback between the heating and material strength. Equations for benchmark test I and the detailed discussion are included in section 'Benchmark Test I'.

4.2. Ogawa's Model

[21] In this section, we compare the fourth-order Runge-Kutta explicit method with the second-order central difference and full Newtonian implicit schemes employed by ABAQUS for one-dimensional shear heating case. Figure 2a is the schematic diagram for the original model by Ogawa [1987] where the cause of deep focus earthquake was explored as a result of shear instability within a subducting viscoelastic lithosphere. Stress- and temperature-dependent rheology was also assumed, and the deformation rate was set at a constant strain rate of 10−13 s−1. In addition, the magnitude of initial stress and temperature were prescribed as 400 MPa and 978 K, respectively. We have assigned the temperature perturbation around the lower boundary with a small amount of 10 K and a length scale of 0.1 km. This perturbation is applied to promote shear localization. Additional information for this benchmark test is in Tables 1 and 2. By demonstrating that the temperature in the shear zone can rise quickly up to additional 100–400 K, Ogawa [1987] concluded that shear heating could be a viable mechanism for triggering deep focus earthquakes.

Figure 2.

(a) The schematic description for Ogawa's [1987] model. Temporal temperature variation on homogeneous viscoelastic material is integrated under constant shearing rate condition. Simple one-dimensional evolution (z axis versus temperature) is calculated. (b) Temperature evolution with time at the central node of domain. Red and blue lines show temperature evolution from the explicit and implicit schemes, respectively. The explicit scheme makes faster and stronger shear instability. (c) The temperature profile at time = 400 Myr. Red and blue lines refer temperature profiles using the explicit and implicit schemes, respectively.

[22] We report here the numerical experiment of Ogawa [1987], using both schemes. Figure 2b is the comparison among different approaches. The explicit scheme is much closer to the result by Ogawa [1987]. Furthermore, the temperature evolutions at the shear zone predicted by the explicit and implicit schemes are different. In the early stage of viscous dissipation, the explicit and implicit schemes produce relatively similar outcome. However, in the latter stage, temperature elevation in the explicit scheme is much more rapid. Temperature in the shear zone rises faster and faster with time, due to the one-dimensional nature that limited diffusion [Ogawa, 1987]. The explicit scheme is appropriate for calculating the shear instability propagation which has a similar time scale with time steps of the scheme [Hulbert and Chung, 1996]. Therefore, the curves of the explicit and implicit schemes become very different at the latter stage of viscous dissipation.

[23] In Figure 2c, the temperature profiles at time of 400 Myr are obtained by the explicit (red line) and implicit (blue line) schemes. It shows that temperature profiles of depth between 15 and 50 km are almost the same. However, in the region where the shear heating takes place, the temperature profiles are quite different. The explicit scheme produces two times larger temperature increase than that found with the implicit scheme.

[24] A higher shearing rate induces vigorous positive feedback between temperature and plastic strain. If shear heating in Ogawa's model is governed by the quasi-static mechanism which would be well resolved by a large time step, the time step in the implicit scheme should not be fluctuating, rather it should be uniform. Otherwise, the case of higher shearing rate is expected to be highly nonlinear. Therefore, small time stepping is necessary to calculate the dynamic effect properly. Figure 3 illustrates the temporal evolution of time step with different shearing rates under the implicit scheme. Thick and thin lines in Figure 3 depict variations of time step for slow shearing (≤1.5 cm/yr) and fast shearing (≥3 cm/yr), respectively. In the beginning stage, all models show the sharp increasing of time step, because the initial time step is set to be 1 s. For the case of slow shearing, the time step is large and almost uniform throughout the whole time domain. Otherwise, all the thin lines are significantly fluctuating (see yellow stars in Figure 3). The black thin lines (for the case of 9.0 cm/yr) and the blue thin lines (for the case of 3.0 cm/yr) exhibit the fastest and latest fluctuations of time step, respectively. This means that there is a marked correlation between time step and nonlinearity from the fast shearing rate. Intense deformation rates cause large temperature increases. It forces the implicit solver to reduce drastically the time step. Figure 4 shows the total iteration number with varying shearing rates. As expected, higher shearing rates increase the number of iterations necessary for the convergence within a given tolerance of O(10−5).

Figure 3.

Temporal evolution of time step with different shearing rate under the implicit scheme. Thick and thin solid lines depict cases of slow and fast shearing rates, respectively. Yellow stars point out the moment when the explicit scheme with short time stepping should be applied.

Figure 4.

Total number of iterations with varying shearing rates. As expected, the higher shearing rate is applied, the larger number of iterations is required. Many iterations show clearly the long computing time.

5. Two Cases of Shear Zone Development

[25] This section describes the two previous studies of shear zone development in the lithosphere that will be reexamined with our new adaptive scheme (i.e., ATS). The R model involves the case of lithospheric necking [Regenauer-Lieb and Yuen, 1998] and the S model is related with the development of shear zone at the interface of two materials with different elastic shear moduli [So et al., 2012]. Originally, both R and S models employed the implicit scheme.

5.1. R Model

[26] Figure 5a shows the configuration of the model where the lithosphere is 800 km long and 100 km high. The rheology is elastoplastic, that is, the lithosphere behaves elastically below a certain stress criterion, but upon exceeding this yield criterion it deforms plastically. When the domain behaves plastically, the temperature/stress-dependent creep rheology derived from laboratory experiments [Chopra and Paterson, 1981] is assigned (see equation (9)). The lithosphere was extended at a rate of 2–8 cm/yr from the right. The left side is fixed whereas the top and bottom boundaries are prescribed as a free surface. A notch was prescribed at the top of the lithosphere so that the necking would start at that location.

Figure 5.

Descriptions for (a) R and (b) S models. R model uses homogeneous material and is suitable to observe tendency of deformation with three different time-integration schemes because of the notched zone where the stress is extensively concentrated. S model has the domain, composed of two elastically heterogeneous elastoplastic materials (i.e., bimaterial situations).

[27] According to Regenauer-Lieb and Yuen [1998], the plastic yielding starts at around 0.725 Myr, and within the next 0.1 Myr, the shear instability propagates until it reaches the base of the lithosphere. Once this point is reached, the temperature field of the lithosphere gradually becomes stabilized as the shear heating generated at the shear zone is balanced by the thermal diffusion toward the surrounding lithosphere whose temperature is relatively low.

5.2. S Model

[28] Figure 5b describes the configuration of the model examining the generation of asymmetric instability zone at the interface of two materials with different shear moduli. The lithosphere is 600 km long and 150 km high with an elastoplastic rheology. The boundary conditions assigned to this model is similar to those of R model. However, unlike Regenauer-Lieb and Yuen [1998], a weak zone such as fault and low viscosity zones was not predefined. The experiment was performed with shear modulus contrast of 3 and constant compression rate of 2 cm/yr.

6. Results

[29] Figure 6 is the plot of temperature field of the R model generated using the ATS. The implicit scheme was used for the stage 1 before plastic yielding and the stage 3 which corresponds to a postshear-zone-development period where the heat generated at the shear zone is balanced by thermal diffusion. The explicit scheme was used for the stage 2 of plastic yielding, the initiation of shear instability and its rapid propagation. These results were compared with those obtained using single time stepping approaches (i.e., both implicit and explicit schemes). The overall pattern of shear heating and deformation was not much different among the different schemes. However, around the notch where plastic yielding and shear instability occur, a notable difference can be discerned among the predictions.

Figure 6.

Numerical results of R model using the ATS. Detailed explanation of each stage, as given in Figure 1, is consistent with this figure.

[30] Figures 7a and 7b show respectively deformation around the notched area and the temperature field for different schemes. The final shape of the notch is not much different from its original configuration in the case of the implicit scheme, but it is more accentuated and localized for the explicit and the ATS (Figure 7a). The temperature at the notch becomes higher, as one changes the time stepping method from the implicit and explicit methods to the ATS. The difference in resulting temperature at the notch can be manifested more clearly in Figure 7a which shows that not only does the temperature becomes higher but also is more confined for the ATS than for the other two schemes.

Figure 7.

(a) The deformation appearances and temperature distributions around notched zone at 1 Myr of R model. The deformation of the ATS produces the sharpest compared with other schemes. (b) Temperature profile along the notched zone at 1 Myr of R model. Consistent with Figure 7a, the ATS displays the most localized and highest temperature field.

[31] The emergence of fine-scale features when employing the ATS is also evident in the S model. Figure 8 is the temperature profile at the interface between two different parts of lithosphere. Again, a higher and more localized temperature is found around the asymmetric shear zone when using the ATS than in the single time-stepping schemes.

Figure 8.

The temperature profiles along interface at 1 Myr with three different schemes for S model. The width of shear localized zone with the ATS is almost two times more localized compared with the implicit case.

[32] Another important benefit of using the ATS is that the solution is stable and convergent over a wider range of parameters. For instance, if one uses the implicit scheme alone, the solution diverges with increasing rate of extension in the R model. This shortcoming is demonstrated in Figure 9 where the red star symbols represent the time beyond which the implicit solver fails due to growing nonlinearity. The time span during which nondivergent solution can be obtained becomes shorter with increasing extension rate. The implicit solver tries to circumvent this problem by reducing the time step near the plastic yielding point but there is a limit to the reduction ensuring numerical convergence. As a result, the implicit method cannot handle cases with extremely high strain rates.

Figure 9.

The variation of time step with higher extension rate under the implicit scheme. Higher deformation rate induces quicker divergence. To reach convergence, we must use an explicit scheme at the time of divergence in the solution, as indicated by the red stars.

[33] An important issue in using the ATS is when to change between the different schemes. Figure 10a is an example of the variations in time step size for the implicit scheme. In most routines, this time step size is automatically adjusted based on a certain criterion and tests performed after each iteration. In the case of R model, the time step size is reduced from ∼30,000 to ∼3000 years (Figure 10a). However, such a brute-force tactic involving a simple reduction may not be sufficient to guarantee the accuracy of the solution near the shear zone.

Figure 10.

(a) The evolution of time step of R model in the implicit scheme with a slow extension of 2 cm/yr. The almost uniform time step is shown, but clear time step reduction in the stage of shear instability initiation appears. Higher velocity than 2 cm/yr does not converge in the implicit scheme. (b) Temporal variation of time step in R model under the ATS with extensional velocity = 2 cm/yr. Blue and red regions represent respectively implicit and explicit stages. The temporal evolution of time steps in S model with the ATS displays a similar trend with R model.

[34] According to So et al. [2012], the velocity of shear instability propagation is ∼5·10−6 m/s which differs greatly from the deformation rate of ∼5·10−10 m/s during stage 2. It may be more practical to switch from the implicit to explicit schemes, in order to catch the detailed features with much finer time steps. In our study, we reduced the time step size down to 0.3–3 years for stage 2 of the ATS in R model (Figure 10b).

[35] The disadvantage of the explicit method is that it takes an immense amount of computational time due to its extremely small time stepping. Therefore, when the balance is reached as in stage 3, it may be better to switch back to the implicit scheme. However, unlike the transition from stages 1 to 2, the exact timing may not be clear for transition from stage 2 to 3 since the thermal balance is reached slowly over a long time scale.

[36] Figure 10b shows the variation in the time step for the ATS. The first and last stages, which are covered by the implicit scheme, represent respectively the elastic energy storage and maturing of shear localization. In Figure 10b, sharp time step change appears at the beginning step of the implicit scheme employed stages (see blue regions). The implicit scheme solver quickly increases time step from 1 to 1011−1012 s (3200–32,000 years) to save computational cost, since solutions in this stage are expected to be relatively stable. In the middle stage (red region), highly nonlinearity, sharp temperature change and fast shear instability propagation demanded the explicit scheme with small time step. The implicit scheme can handle the small time step, but the size of tangent stiffness matrix becomes larger with smaller time step and the convergence may not be guaranteed within a given tolerance O(10−5) [Choi et al., 2002]. We can definitely keep adopting purely implicit scheme for all stages 1, 2, and 3 with relaxing the error tolerance for using small time stepping. If we relax the tolerance and then use small time stepping in purely implicit scheme, the accuracy is not guaranteed. Otherwise, if we use small time stepping with a fixed tolerance of 10−5, the calculation of tangent stiffness matrix may be diverged. The way to get around this contradiction is to develop a new robust implicit solver. If the implicit solver is improved, we can catch the small time stepping for short timescale physics and the accuracy with tight tolerance. This effort has been paid in applied mathematics [e.g., Burckhardt et al., 2009]. On the other hand, the explicit scheme can manage very short time scale and highly nonlinearity and a large stiffness in time without any problems for convergence. In red region in Figure 10b, the time step hovers around 107−108 s (0.3–3 years). This time step is consistent with that of Ogawa's [1987] study, which argued that the time scale for earthquake phenomenon is reduced from few hundreds of years to few years. This extremely short time step does not cause any problems with convergence and allows an accurate determination of temperature elevation (∼11 K at 1 Myr) from elastic energy release whose characteristic time scale is very short. The evolution of time steps in S model with the ATS displays a similar trend with the R model.

7. Discussion

[37] We have studied the effectiveness of the ATS method for modeling geodynamical phenomena with multiple time scales. In particular, we have compared the single time stepping methods against the ATS for shear zone development in the lithosphere by extension or compression.

[38] The process of shear zone development can be divided into three stages. The stage 1 can be regarded as a period where the stress inside the medium builds up. The response of the lithosphere to the external force is purely elastic. The gradual buildup of stress eventually causes plastic yielding, and the stage 2 can be defined as the start of the yielding. An important aspect of the plastic yielding is the generation of shear heating which in turn reduces the mechanical strength. Although the amount may be small at the beginning, the temperature around the shear zone can grow rapidly as a result of a positive feedback between shear heating and deformation. The stage 2 thus can be characterized as a period during which the shear zone develops and the instability occurs within the lithosphere. In the case of R model, the yielding zone eventually reaches the bottom of the lithosphere. In the S model, two yielding zones, one at the top and the other at the bottom, merges at the center of the lithosphere. Then, the temperature within the lithosphere increases steadily as a balance is reached between the heat generated by the shear deformation and outward heat diffusion. The time at which this equilibrium occurs can be defined as the beginning of the stage 3. Unlike an abrupt transition from the stages 1 to 2, the transition from the stages 2 to 3 may be gradual. In the implicit scheme, the transition between the different stages can be clearly identified by the change in time stepping sizes.

[39] Before investigating the benefits of the ATS, it is important to explore the characteristics and reliability of our technique. Two benchmark tests were performed. In the benchmark test I, the implicit and explicit schemes were compared for a case involving shear heating within pure viscous fluid. The test was done because an analytical solution exists in this particular case and also the mathematical description of the problem is somewhat similar to that of lithospheric shear zone development. In Ogawa's model, where the explicit method of Ogawa [1987] was compared with our results, the two numerical results were almost identical except for a slight difference in the initiation time of instability.

[40] When dealing with multitimescale physics, it is important to understand how the differences in time scales may affect the numerical solutions. One way to characterize fast and slow processes is to compare the characteristic velocities of the different components in the computational domain. In the stage 1, where the lithosphere responds elastically according to the prescribed boundary condition, the characteristic velocity can be equated to the deformation rate. On the other hand, in the stage 2, where the important change in the system is caused by the reduction in mechanical strength as a result of plastic yielding and shear heating, the characteristic velocity can be regarded as the propagation speed of the shear zone. For the stage 1, the characteristic velocity is approximately 10−10 m/s. Compared to that of the stage 2 where the estimated propagation speed is roughly 10−6 m/s, the extraordinary characteristic velocity of stage 2 is 104 times faster but the duration of this period is extremely short. Unfortunately, most implicit schemes cannot adjust to such a rapid change in time step sizes within a given tolerance of error O(10−5) between the successive iterations. A viable option is to change from the implicit to the explicit schemes.

[41] The abrupt change of the inertial term is important for controlling the whole crustal system, especially for slow earthquakes [Homburg, 2013]. However, the deformation rate is continuous during the transition between stages 1 and 2. This means that the inertial term is nearly zero during our run. However, the propagation of thermal instability occurs suddenly and is very fast. This is the reason why we adopted the explicit scheme only for the energy equation.

[42] Comparison between the ATS and single time stepping schemes results shows that, although the overall features appear to be similar, one important advantage gained by the ATS is that it produces a much fine-scale image near the localized shear zone (see Figures 7 and 8). For instance, according to the ATS, the temperature at the shear zone becomes higher by a few degrees after 1 Myr the deformation field becomes more localized.

[43] The ability to predict correctly deformation and temperature at the shear zone has important implications for understanding the nature and behavior of abrupt feature such as fault, subduction initiation and slab detachment. Great efforts have been made to fill the gap between the tectonic stress on the lithosphere (e.g., slab pull and ridge push) and the required stress for explaining observed large ruptures on the Earth. The stress found in the nature is much smaller than the required stress for the shear deformation in the field [Regenauer-Lieb et al., 2008], thus many complex mechanisms, such as grain-size related weakening [Yamasaki, 2004] or damaged rheology [Karrech et al., 2011], have been employed. In summary, our ATS has the same aim of reducing the stress for explaining observed large deformation and shear heating, but we take a distinct approach of choosing the ATS, which can help in the re-evaluation of the immediate deformation history in terms of choosing the proper time-integration scheme.

[44] Moreover, one of the most intensively debated issues is related with the strength of fault and timing at which faults will slip. Predicting the exact temperature at the fault zone is vital as temperature is one of the major factors controlling the effective strength of the material. Our study, which shows that the pattern of temperature distribution at the fault zone can be quite different, depending on whether the ATS or single time-stepping approach is employed, can have an important impact on the studies involving local ductile instabilities, which may cause earthquakes [Ide et al., 2007].

8. Conclusions

[45] We have highlighted the importance of selecting appropriate schemes (i.e., implicit, explicit, the ATS schemes) for strongly nonlinear and stiff geodynamical problems, which have multiple time scales. We find that stages of triggering of elastic energy release and geometrical failure would require very small time steps. Otherwise, relatively large time step is enough for resolving steady phenomena, such as the stage before plastic yielding and the stage after achieving the equilibrium between shear heating and thermal diffusion. This strategy of choosing judiciously short and long time steps should be for solving geodynamical problems and thus the modeler should consider the ATS and test the timing of switching from one into the other. The suitable schemes for the proper physical situations have been insinuated by King [2008], Regenauer-Lieb and Yuen [1998], and Kassam and Trefethen [2005], who noted that direct-explicit and iterative-implicit schemes should be chosen according to the characteristic of solution of problems. If a steady solution is not expected, the explicit scheme is better. Within this context, reliable previous studies, R and S models, have been comparatively analyzed with considering three different schemes. Generally, in two-dimensional models, the ATS reveals the strongest and the most localized instability, which is favor to lithosphere-scale instabilities. Using either implicit or explicit schemes leads to underestimate the temperature elevation (∼4 K at 1 Myr in the R and S models and ∼200 K at 400 Myr in the Ogawa's model) by shear instability. Moreover, the implicit scheme has the tendency that the solution blows up, when strong nonlinearity is involved (e.g., fast deformation rate), because the solver uses short time stepping, causing rapidly growing of tangent stiffness matrix under a given error tolerance [Choi et al., 2002]. On the other hand, the extremely short time step in the explicit scheme requires too many time steps. Therefore, the explicit scheme does not efficiently calculate elastic energy storage in quasi-static compressional or extensional stages. Explicit and implicit methods have both advantages and drawbacks in terms of the computational time and the ability to capture the proper time scales. Therefore, we can advocate that this adaptive time-stepping (i.e., ATS) strategy can be helpful to other challenging problems in geodynamics, such as magma and mantle dynamics with multiple time scales.

Appendix: A

A1. Explicit Scheme

[46] The explicit time-stepping scheme for thermal analysis in ABAQUS adopts the second-order central difference integration rules [Hibbit, Karlsson and Sorenson Inc., 2009]:

display math(A1)
display math(A2)
display math(A3)

where C is the diagonal lumped capacitance matrix [Pham, 1986], Pand F are the applied nodal source and internal flux vectors, respectively, k means the index of time step and inline image are midincrement indexes. inline image and inline image definitely refer the first and second time derivatives of T, respectively. Δtk is the time step at index k. The time step in the explicit scheme for convergence and accuracy should be selected more carefully than that for the implicit scheme. The time step Δt for the convergence is theoretically and practically defined according to the Hibbit, Karlsson and Sorenson Inc. [2009]:

display math(A4)

where Lmin is the spatial scale of the smallest element. α represents thermal diffusivity,

display math(A5)

k and cp are thermal conductivity and specific heat, respectively. For strong viscous heating, the time steps are controlled by the fastest rate of the frictional heat production.

A2. Implicit Scheme

[47] ABAQUS/Standard (Implicit) uses full Newton-Raphson iterative solver [Axelsson, 1977]. For mechanical modeling, simple procedure description at time = t is below:

display math(A6)
display math(A7)

[48] Index i represents ith iteration. inline image means displacement vector of ith iteration, respectively. inline image is tangent stiffness matrix in configuration at time = t. inline image and inline image refer the applied force and internal force vectors at t and ith iteration, respectively [Sun et al., 2000]. inline image, inline image and inline image are calculated from inline image and then we can compute inline image using equation (A7). inline image is derived by solving equation (A6) and then this iteration will be repeated until the convergence of inline image is ensured.

[49] For thermal modeling which calculates inline image (temperature of ith iteration at time = t), inline image, inline image and inline image are replaced into the conductance matrix (i.e., inline image), applied nodal heat source (i.e., inline image) and internal heat flux (i.e., inline image) vectors, respectively.

[50] For dynamic time integration, the implicit scheme applies first-order backward Euler operator and Hilber-Hughes-Taylor [Hughes et al., 1977] method which weights to information at t and t + Δt:

display math(A8)
display math(A9)
display math(A10)

where inline image

[51] M is mass matrix. K is tangent stiffness and conductance matrixes in mechanical and thermal modeling. inline image and inline image are the first and second time derivatives of u (displacement and temperature for mechanical and thermal modeling, respectively). α definitely should be a negative real number. α = −0.05 is selected as a default value in ABAQUS [Hibbit, Karlsson and Sorenson Inc., 2009]. For problems with strong shear heating, the time steps are controlled by the maximum rate of heat production.

A3. Benchmark Test I

[52] Previous studies dealing with comparisons between experimental results and numerical simulations [Nezo et al., 2011; Noels et al., 2004] have demonstrated that the explicit scheme is more similar to experimental results when the time scale of the physics is short. Since the laboratory experiments of geodynamical situations with multiple scales are technically very difficult, we need to benchmark both schemes with the analytical solution for shear heating within the lithosphere with stress- and temperature-dependent rheology. Few people have focused on comparing their numerical results with the analytical solution because it is not easy to derive the solution of the problem due to the coupling between complicated rheology and governing equations.

[53] In a theoretical study done 40 years ago, Sukanek et al. [1973] found an analytical solution of shear heating in viscous fluid with the temperature-dependent viscosity (μ(T), equation (A11)). Turcotte and Schubert [2002] extended the solution to the geological scale and solved the differential equation, which is composed of the Brinkman number (Br, equation (A12)) and a dimensionless temperature (θ, equation (A13)) with a simplified assumption consisting of a constant tangential stress τ on the top and uniform initial temperature T0 of the whole one-dimensional fluid with h = 100 km depth. The values of thermal conductivity (i.e., k) and activation energy (i.e., Q) are 4 W/(m·K) and 400 kJ/mol, respectively. R is universal gas constant. Equation (A14) is the analytical relationship between the maximum temperature and the Br number [see details in section 7-5 of Turcotte and Schubert 2002]. We have compared the solution with our steady state results from the one-dimensional explicit and implicit schemes with the same mesh size (∼0.05 km, ∼2000 grid points). Time steps for each scheme are automatically determined by system configuration of each time step.

display math(A11)
display math(A12)
display math(A13)
display math(A14)

[54] The quasi-static stage for elastic energy storing is not necessary for the purely viscous, fluid-like, viscoelastic model. Thus, we do not need to use the ATS. The purpose of this benchmarking is to determine which scheme is more appropriate for solving shear heating with a short time scale and complicated feedback between the mechanical and thermal instabilities. The Br number is changed along the branch only by varying the tangential stress for a given initial temperature of T0 = 400 K and 1000 K. A low T0 leads to a subcritical branch in which the initial viscosity is large, and thus, the tangential stress is the dominant factor for the temperature elevation. Otherwise, a high T0 induces the supercritical branch in which the initial viscosity is low and the stress cannot be transmitted to the bottom. In this branch, the viscosity should be decreased for a mechanically and thermally steady state even in the case of large tangential stress.

[55] On the subcritical branch, the difference between the results from the low and high Br numbers is obvious (see the blue region in Figure 11). With the low Br number, both schemes generate a similar temperature because of the weak nonlinearity due to a low tangential stress leading to a small variation in the rheology and temperature. On the other hand, the explicit scheme shows far greater accuracy to the analytical solution and a high Br number. High stress induces a sharp change in the temperature and rheology, and this high nonlinearity requires short time stepping with the explicit scheme. In the case of the supercritical branch (see the red region in Figure 11), both schemes show completely distinct results. This branch hardly converges to the steady state due to the low viscosity and the subsequent high strain rate with the implicit scheme. This result is consistent with our results showing that the implicit solver cannot guarantee numerical convergence within a given tolerance (10−5) when a too high a deformation rates are assigned. Otherwise, the maximum temperatures from the explicit scheme are close to the analytical solution. In this section, we have confirmed again the usefulness of the explicit scheme for solving problems involving shear heating in situations with simultaneous high shear stress and strain rate.

Figure 11.

The comparison between analytical solution and numerical result with the explicit and implicit schemes. Black solid curve represents the analytical solution. Open rectangles and circles show numerical solution with the implicit and explicit schemes. Generally, the explicit scheme is more accurate to the analytical solution.


[56] This research has been supported by Brain Korea 21 Project and Korean government (MEST, 2009–0092790) for B.-D. So and S.-M. Lee and Foundation grant in Collaboration of Mathematics and Geosciences (CMG) program for D. A. Yuen. We would like to thank J. Brown for his online conversations, which have led to this work and M. Knepley for constant encouragement.