An implicit free surface algorithm for geodynamical simulations

Abstract

Identifying the dominant controls on Earth's surface topography is of critical importance to understanding both the short-and long-term evolution of geological processes and past-and present-day dynamics of Earth's coupled mantle-lithosphere system. The ability to simulate a stress free -or a so-called 'free surface' -boundary condition is required to examine such processes via numerical models. However, at present, geodynamical models incorporating a free surface are limited, as most underlying free surface algorithms place severe restrictions on the computational timestep. Consequently, the simulations are often intractable. In this study, we introduce a new approach for incorporating a free surface within geodynamical models: an algorithm, in which free surface elevation is treated as an independent variable and is solved for in conjunction with the momentum and continuity equation, using implicit time integration. We demonstrate that the method is straightforward to implement in existing models and, using a series of analytical and benchmark comparisons, we show that it does not suffer from the timestep constraints of previous schemes. Furthermore, the scheme can be made second order accurate in time, at no additional cost. The method therefore dramatically improves the computational efficiency of geodynamical simulations including a free surface, whilst maintaining solution accuracy.

A hierarchical mesh refinement technique for global 3-D spherical mantle convection modelling

“…Consequently, although the scheme may be less beneficial than the fully adaptive techniques currently under development (e.g. Davies et al, 2011;Kronbichler et al, 2012;Burstedde et al, 2013), it will allow pre-existing codes to examine more challenging problems than have previously been possible (indeed, it has already done so, as demonstrated by Davies and Davies, 2009). Given the amount of investment that has gone into these codes, a method such as that presented, which will extend their lifetime and applicability, is a worthwhile and significant development.…”

“…Davies et al, 2007Davies et al, , 2008Stadler et al, 2010;Leng and Zhong, 2011), leading to the development of several state-of-the-art computational frameworks for simulating global mantle convection. The most notable examples are: (i) Fluidity Kramer et al, 2012); (ii) ASPECT (Kronbichler et al, 2012); and (iii) RHEA (Stadler et al, 2010;Burstedde et al, 2013). Such codes, which employ cutting-edge methods in mesh refinement, solver technology and parallelisation, open up a whole new class of problems for mantle dynamics research, as demonstrated by Stadler et al (2010).…”

Lithospheric Thinning by Mantle Plumes

“…We solve the Stokes and energy equations relevant to mantle convection using Fluidity, a finite-element, control-volume model with several features that place it at the forefront of computational fluid dynamics (Davies et al, 2011;Kramer et al, 2012). Fluidity: (i) uses an unstructured mesh, which enables the straightforward representation of complex geometries; (ii) dynamically optimizes this mesh, providing increased resolution in areas of dynamic importance, thus allowing for accurate simulations across a range of lengthscales, within a single model; (iii) enhances mesh optimization using anisotropic elements (Davies et al 2011); (iv) is optimized to run on parallel processors and has the ability to perform parallel mesh adaptivity; (v) utilises solvers linked to PETSc (Balay et al 1997), that can handle sharp, orders of magnitude variations in viscosity, a feature that is essential for the proposed study; and (vi) employs a free-surface at the outer domain boundary (Kramer et al 2012).…”

“…Fluidity: (i) uses an unstructured mesh, which enables the straightforward representation of complex geometries; (ii) dynamically optimizes this mesh, providing increased resolution in areas of dynamic importance, thus allowing for accurate simulations across a range of lengthscales, within a single model; (iii) enhances mesh optimization using anisotropic elements (Davies et al 2011); (iv) is optimized to run on parallel processors and has the ability to perform parallel mesh adaptivity; (v) utilises solvers linked to PETSc (Balay et al 1997), that can handle sharp, orders of magnitude variations in viscosity, a feature that is essential for the proposed study; and (vi) employs a free-surface at the outer domain boundary (Kramer et al 2012). Fluidity, therefore, provide a base for accurately and efficiently simulating plume-lithosphere interaction under Earth-like conditions (see Figure 1).…”

Implicit solution of the material transport in Stokes flow simulation: Toward thermal convection simulation surrounded by free surface

“…Such schemes are conditionally stable, and thus an inappropriate choice of the computation time step will result in spurious behavior, which typically manifests itself as an oscillation at the free surface interface [3,12,13]. The state of pressure within the Earth is dominated by a large background hydrostatic pressure.…”