Grid convergence study of the combined finite difference & Marker-In-Cell method for geodynamic applications

The combined finite difference & Marker-In-Cell technique of Gerya and Yuen (2003) is a popular method for studying geodynamic processes involving many material phases with differing rheological properties. Numerous applied studies have demonstrated the versatility of the scheme for studying large viscous deformations. The principal reason why the method is flexible is the use of an interpolation technique to capture fluid properties at a subgrid level. Given the geodynamic applications, the discretisation is required to be robust when the domain contains sharp variations in viscosity and free surfaces. In our study, we measure the robustness of the spatial discretisation by computing the discretisation error and analysing the convergence rate of the velocity and pressure fields. We examine the convergence properties of the traditional staggered grid discretisation and show that introducing an interpolated viscosity constructed from the markers does not modify the convergence behaviour. Moreover, we introduce a free surface stabilisation algorithm and show that the stabilisation does not alter the convergence rate of the method. Gerya, T. V., Yuen, D. A., 2003. Charaterictics-based marker method with conservative[|#12#|]finite-difference schemes for modeling geological fows with strongly variable transport properties. Physics of the Earth and Planetary Interiors 140 (4), 293-318.