Large time-step numerical modelling of the flow of Maxwell materials

Maxwell viscoelastic materials are commonly simulated numeri-cally in order to model the stresses and deformations associated with large-scale earth processes, such as mantle convection or crustal de-formation. Both implicit and explicit time-marching methods require that the timesteps used be small compared with the Maxwell relax-ation time if accurate solutions are to be obtained. For crustal tec-tonic modelling, where Maxwell times in a ductile lower crust may be of order of a decade or less, the large number of timesteps required to model processes lasting many millions of years imposes a huge com-putational burden. This burden is avoidable. In this paper I show that, with the appropriate formulation of the problem, timesteps may be taken which are much larger than the Maxwell time without loss of accuracy, as long as they are not large compared with the inverse strain rates (“tectonic ” timescales) in the model. The method relies on explicit analytic integration of the Maxwell constitutive relation 1 for the stress over time intervals which may be longer than the re-laxation time as long as they are short compared with the timescale over which crustal stresses and geometries change. The validity of the formulation is also demonstrated numerically with two simple models, one involving extension of a uniform block, and the other involving the shear of a composite layer