EGU General Assembly 2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.

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

Adina E. Pusok, Dave A. May, and Richard F. Katz
Adina E. Pusok et al.
  • University of Oxford, Department of Earth Sciences, Oxford, United Kingdom of Great Britain and Northern Ireland (

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.

How to cite: Pusok, A. E., May, D. A., and Katz, R. F.: Magma dynamics using FD-PDE: a new, PETSc-based, finite-difference staggered-grid framework for solving partial differential equations, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-18690,, 2020


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  • CC1: Comment on EGU2020-18690, Lawrence Hongliang Wang, 03 May 2020

    Cool project!  I wonder how easy it is to create FD-PDE project and change equations

    • AC1: Reply to CC1, Adina E. Pusok, 04 May 2020

      Thanks! Provided that you have your model problem defined and are comfortable with PETSc, then setting up an application is quite easy. We have already many examples and tests on how to use the FD-PDE framework that should be helpful to someone new to it. Of course, the challenge will be to build complex science applications, and we are working on some new mid-ocean ridge models.