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

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.