Large time step numerical modelling of the flow of Maxwell materials

Abstract

SUMMARY Maxwell viscoelastic materials are commonly simulated numerically in order to model the stresses and deformations associated with large-scale earth processes, such as mantle convection or crustal deformation. Both implicit and explicit time-marching methods require that the time steps used be small compared with the Maxwell relaxation time if accurate solutions are to be obtained. For crustal tectonic modelling, where Maxwell times in a ductile lower crust may be of order of a decade or less, the large number of time steps required to model processes lasting many millions of years imposes a huge computational burden. This burden is avoidable. In this paper I show that, with the appropriate formulation of the problem, time steps 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 times over which strain rates vary significantly (‘tectonic’ timescales) in the model. The method relies on explicit analytic integration of the Maxwell constitutive relation for the stress over time intervals, which may be longer than the relaxation 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 by comparison with the analytic solutions for three simple plane-strain models: extension of a uniform block, shear of a composite layer and development of a Rayleigh‐Taylor instability.