Robust computational techniques for studying Stokes flow within deformable domains: Applications to global scale planetary formation processes

Abstract:

We develop numerical schemes for solving global scale Stokes flow systems employing the “sticky air” (approximate free surface) boundary condition. Our target application considers the dynamics of planetary growth involving long time-scale global core formation process, for which the interaction between the surface geometry and interior dynamics play an important role (e.g. Golabek et.al. 2009, Lin et.al. 2009). The solution of Stokes flow problems including a free surface is one of the grand challenges of computational geodynamics due to the numerical instability arising at the deformable surface (e.g. Kaus et.al. 2010, Duretz et.al. 2011, Kramer et.al. 2012).

Here, we present two strategies for the efficient solution of the Stokes flow system using the “spherical Cartesian” approach (Gerya and Yuen 2007). The first technique addresses the robustness of the Stokes flow with respect to the viscosity jump arising between the sticky air and planetary body (e.g. Furuichi et.al. 2009). For this we employ preconditioned iterative solvers utilising mixed precision arithmetic (Furuich et.al. 2011). Our numerical experiment shows that the mixed precision approach using double-double precision arithmetic improves convergence of the Krylov solver w.r.t to increasing viscosity jump without significantly increasing the calculation time (~20%). The second strategy introduces an implicit advection scheme for the stable time integration for the deformable free surface. The Stokes flow system becomes numerically stiff when the dynamical time scale associated with the surface deformation is very short in comparison to the time scale associated with other physical process, such as thermal convection. In this work, we propose to treat the advection as a coordinate non-linearity coupled to the momentum equation, thereby defining a fully implicit time integration scheme. Such an integrator scheme permits large time step size to be used without introducing spurious numerical oscillations. The non-linear equations defined by the implicit time integration scheme are solved using a JFNK framework (Knoll and Keyes 2004). We have conducted a series of numerical experiments which clarify the trade-off’s between the computational cost associated with solving the non-linear problem and time step size.