Adaptive Mesh Refinement for Multiscale

plied and theoretical physics is to fully understand and predict the behavior of systems far from thermodynamic equilibrium,1–4 including those systems driven by an external force or experiencing a sudden change in environment (such as pressure or temperature). They also include systems transitioning from one metastable or long-lived state to another. The need to accurately model and numerically simulate these processes has become more pressing as efforts to derive wellparameterized evolution equations for mesoscopic nonequilibrium dynamical processes’ progress. Simultaneously, it’s becoming increasingly important (for example, in phase transition kinetics5,6) to be able to validate equations and their solutions against experiment. This validation requires highly accurate and efficient partial differential equation (PDE) solvers. PDE integrators for these problems aren’t typically selected for their space and time accuracy. Instead, researchers have used easily implemented, mostly explicit methods to determine universal features such as domain growth exponents. For modern materials applications, however, space and time resolution are crucial for modeling system behavior. For these applications, the standard low-order or fixed-grid integration approach is prohibitive due to time-step constraints. We address this issue by applying adaptive mesh refinement (AMR) methods in dynamical condensed-matter systems. For ease of exposition, we focus on the time-dependent (real) Ginzburg-Landau (TDGL) equations because they’re the prototypical example of a spatially extended nonequilibrium system undergoing a dynamical phase transition.