Magma dynamics using FD-PDE: a new, PETSc-based, finite-difference staggered-grid framework for solving partial differential equations

All divergent plate boundaries are associated with magmatism, yet its role in their dynamics and deformation is not known. The RIFT-O-MAT project seeks to understand how magmatism promotes and shapes rifts in continental and oceanic lithosphere by using models that build upon the two-phase flow theory of magma/rock interaction. Numerical models of magma segregation from partially molten rocks are usually based on a system of equations for conservation of mass, momentum and energy. One key challenge of these problems is to compute a mass-conservative flow field that is suitable for advecting thermochemically active material that feeds back on the flow. This feedback tends to destabilise the coupled mechanics+thermochemical solver. 

Staggered grid finite-volume/difference methods are: mimetic (i.e., discrete differential operators mimic the properties of the continuous differential operators); conservative by construction; inf-sup stable and "light weight" (small stencil) thus they are well suited to address these problems. We present a new framework for finite difference staggered grids for solving partial differential equations (FD-PDE) that allows testable and extensible code for single-/two-phase flow magma dynamics. We build the framework using PETSc (Balay et al., 2019) and make use of the new features for staggered grids, such as DMStag. The aim is to separate the user input from the discretization of governing equations, allow for extensible development, and implement a robust framework for testing. Any customized applications can be created easily, without interfering with previous work or tests.

Here, we present benchmark and performance results with our new FD-PDE framework. In particular, we focus on preliminary results of a two-phase flow mid-ocean ridge (MOR) model with a free surface and extensional boundary conditions. We compare flow calculations with previous work on MORs that either employed two-phase flow dynamics with kinematic boundary conditions (i.e., corner flow, Spiegelman and McKenzie, 1987), or single-phase flow dynamics with free surface (i.e., Behn and Ito, 2008). In the latter case, the effect of magma is parameterised according to a priori expectations of its role. 

 

Balay et al. (2019), PETSc Users Manual, ANL-95/11 - Revision 3.12, 2019.

Spiegelman and McKenzie (1987), EPSL, 83 (1-4), 137-152.

Behn and Ito (2008), Geochem. Geophys. Geosyst., 9, Q08O10.