Unconventional Discretisations of the Temperature Equation in Earth Mantle Dynamics

by Maximilian Volkan Baloglu and co-authors Mohammad Farahani, Christoph Rettinger, Constantin Vogel (29.10.2014)

**The Earth system as we know is highly influenced by the huge forces that arise from mantle convection, which induces the driving forces that can be recognised as plate tectonics, earthquakes or mountain building activities. The numerical simulation of this procedure with a fine enough spatial and temporal resolution is a grand challenge in computational modelling. In this BGCE Honours Project, the focus rests on one single part of the whole picture, namely the temperature advection-diffusion equation. There, we tried to develop and evaluate unconventional discretisations of the temperature equation that tackle the problem of artificial diffusion.**

By modelling the mantel convection, several problems occur while extremely large grids combined with many time steps are required to enable reliable Earth-like physical simulations. This Honours Project is related to the Terra-Neo Project, which is part of the German Priority Programme 1648 to develop software for exa-scale computing and focuses on this problem. It is a joint venture between the geophysicists of the LMU, the computer scientists of the FAU and the mathematicians of the TUM. The overall goal of this interdisciplinary project is to model and simulate the Earth mantle with a spatial resolution which is only possible with exa-scale systems that will be available in the near future. Then this software will allow geophysicists to quantify geophysical phenomena much more detailed than ever before. To solve the underlying Stokes equations, that describe the dynamics of the Earth mantle, the Galerkin Finite Element Method (FEM) combined with a multigrid algorithm is used to achieve the necessary performance.

By trying to solve the Stokes equations with FEM, instabilities arise that require adding artificial diffusion in order to stabilise the procedure. Unfortunately, this approach disturbs and consequently falsifies the obtained solution significantly. Instead of the physically expected thin flow channels, which transport material from the core to the surface of the Earth, one obtains broad channels as a result of the increased diffusion.

In Earth mantle dynamics, it is assumed that very high Péclet numbers occur so that the energy equation can be approximated by a pure advection equation for the temperature. For this purpose, we examined two algorithms in detail, regarding their capability to advect temperature with respect to a given underlying velocity field.

One of them is based on the Bresenham algorithm, which originates from the field of computer graphics and is used there to draw straight lines in a discretised domain. This algorithm is especially designed to maintain sharp edges and thus can be physically interpreted as a diffusion-free transport of a quantity. Unfortunately, simulations reveal inherent shortcomings of the Bresenham Method which could not be overcome by several attempts.

In contrast to the grid-based Bresenham Method, the second algorithm - denoted as Lagrangian Method - makes use of particles for the temperature transport, similar to the algorithm proposed in. These particles do not interact with each other and move through the domain according to the underlying velocity field by carrying the relevant quantities. Since temperature and velocity are nevertheless given on an underlying grid, interpolations between particles and grid vertices are necessary to link Lagragian and Eulerian specification. We extended the Lagrangian Method to work on triangle and tetrahedron grids to offer an interface to the implementation of the Earth mantle dynamics solver of the Terra-Neo project. Additional advantages of the proposed method are the parallelisation capability, a straight-forward extension for advecting other physical quantities and the possible inclusion of physical diffusion in an operator-splitting approach.

**2009-2013**- B.Sc. Computational Engineering, Friedrich-Alexander-University Erlangen-Nuremberg
**Since 2012**- M.Sc. Computational Engineering, Friedrich-Alexander-University Erlangen-Nuremberg
**Since 2013**- Accepted into Bavarian Graduate School of Computational Engineering
**Since 2013**- M.Sc. Scientific Computing (Double master degree), KTH Royal Institute of Technology Stockholm, Sweden