An Exactly Mass Conserving and Pointwise Divergence Free Velocity Method: Application to Compositional Buoyancy Driven Flow Problems in Geodynamics

Abstract

Tracer methods are widespread in computational geodynamics for modeling the advection of chemical data. However, they present certain numerical challenges, especially when used over long periods of simulation time. We address two of these in this work: the necessity for mass conservation of chemical composition fields and the need for the velocity field to be pointwise divergence free to avoid gaps in tracer coverage. We do this by implementing the hybrid discontinuous Galerkin (HDG) finite element (FE) method combined with a mass conserving constrained projection of the tracer data. To demonstrate the efficacy of this system, we compare it to other common FE formulations of the Stokes system and projections of the chemical composition. We provide a reference of the numerical properties and error convergence rates which should be observed by using these various discretization schemes. This serves as a tool for verification of existing or new implementations. We summarize these data in a reproduction of a published Rayleigh‐Taylor instability benchmark, demonstrating the importance of careful choices of appropriate and compatible discretization methods for all aspects of geodynamics simulations.

A Pointwise Conservative Method for Thermochemical Convection Under the Compressible Anelastic Liquid Approximation

“…We found in Sime et al (2021) that precise approximation of the incompressibility, ∇ ⋅ u h = 0, of the velocity field approximation u h , is crucial for precise advection of tracer data. In the compressible scheme we therefore pay close attention to the compressibility approximation ∇ ⋅ u h ≠ 0 error.…”

“…They also offer the tantalizing possibility of exceeding the mesh-dependent Courant limit on time step size. The third major goal of this paper is then to demonstrate the use of the PDE-constrained l 2 projection method to solve Equation 1 at finite κ ϕ instead of κ ϕ = 0 (as in Sime et al, 2021). This also allows us to use tracer methods on the temperature field.…”

“…In our prior work (Sime et al, 2021) we highlighted the necessity for exact satisfaction of the continuity constraint in the incompressible Stokes system for precise advection of tracer data. When combined with a PDE-constrained l 2 projection method (Maljaars et al, 2019) we are also able to exactly conserve mass when advecting chemical data without diffusion.…”

“…The premise of the discretization scheme is to split the advective and diffusive components of the heat equation for individual consideration. For the advection terms we exploit the tracer projection method shown in Sime et al (2021). We then compute the diffusive correction of the projected field followed by updating the tracer data.…”

“…Individual tracers will still converge and diverge from each other in this case (Samuel, 2018) and it is therefore also necessary to have a regular mesh spacing and initialize the tracers at a high enough density to ensure that the number of tracers per cell remains roughly constant. In Sime et al (2021) we demonstrated this using a hybrid discontinuous Galerkin (HDG) finite element (FE) method (Cockburn et al, 2010;Labeur & Wells, 2012;Maljaars et al, 2018;Rhebergen & Wells, 2018a) to discretize the incompressible Stokes equation. The first major goal of this paper is to extend our presentation of the HDG FE discretization of the Stokes equations to a more general and compressible mass-conservation equation and investigate its properties.…”