Overcoming the numerical challenges owing to rapid ductile localization

Abstract. Strain localization is among the most challenging mechanical phenomena for computational Earth sciences. Accurately capturing it is difficult because strain localization initiates spontaneously, is self-accelerating, and its characteristic length and time scales are typically significantly smaller than the spatial and temporal resolutions of the model. This results in an undesirable dependence of the model behavior on numerical parameters and a large computational cost. Strain localization is most commonly associated with brittle failure, but ductile processes such as thermal runaway can also result in rapid ductile localization. Here, we present a numerical model to investigate thermal runaway, and further propose strategies to overcome the challenges associated with resolving rapid localization: (i) adaptive time stepping; (ii) adaptive rescaling; and (iii) two types of regularization. We demonstrate the effect of these strategies in one- and two-dimensional models. We rely on the accelerated pseudo-transient method to solve the governing equations and use graphics processing units to accelerate two-dimensional computations. Our adaptive time stepping strategy allows us to accurately capture spontaneous and rapid stress release during thermal runaway while reducing time steps by more than ten orders of magnitude. Adaptive rescaling further reduces rounding errors and the number of required iterations by two orders of magnitude. Viscosity regularization and gradient regularization enable us to mitigate resolution dependencies but may differently impact the physical response of the model.