Unconventional Discretisations of the Temperature Equation in Earth Mantle Dynamics

Due to the high Péclet-number occurring in Earth mantle dynamics, the energy equation can be approximated by a pure advection equation for the temperature. However, conventional approaches like the Finite Element Method introduce undesired diffusion. Therefore, the task of our BGCE Honours Project is to develop and evaluate unconventional discretisations of the temperature equation that tackle the problem of artificial diffusion. We come up with two different methods: One is based on the Bresenham algorithm stemming from computer graphics. It uses offset values to determine the advection direction of the temperature. The other method features a Lagrangian description of the problem. We achieve this by deploying particles that incorporate temperature values and are advected only by the underlying velocity field. Simulations in one- and two-dimensional domains reveal inherent shortcomings of the Bresenham Method which cannot be overcome by several attempts. On the other hand, the Lagrangian Method yields promising results that can be further improved by additional treatments. We extend 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.