Progress and perspectives on using the Lattice Boltzmann Method for geodynamics simulation research

The Thermal Lattice Boltzmann Method (TLBM) for geodynamical simulation research offers an alternative to classical PDE based methods for 2D and 3D geodynamics simulation research. It is based on modelling the Boltzmann equation on a discrete lattice which involves the movement of number densities carrying mass and energy density on a discrete lattice and their relaxation to equilibrium which model collisions. We present examples in 2D and 3D to illustrate the capabilities, performance, and accuracy of this method for geodynamics research, namely: (1) ability to handle highly nonlinear rheology, ultra-high Rayleigh numbers, a wide range of Prandtl numbers, and multiphase flow, (2) linear scaling up to 300K cores on HPC CPU clusters, and (3) ability to closely match the Blankenbach benchmarks demonstrating the LBMs accuracy. Examples in 2D include high Rayleigh number simulations to Ra = 10¹⁵, highly nonlinear rheology leading to the emergence of plate-tectonic like behaviour, and planetary accretion. Examples in 3D include modelling of a mantle with an aspect ratio of 25x25x1 representing a case from a recent nature paper, and modelling a case of an aspect ratio of 14.4x14.4x1 which is similar that of the Earth for Ra = 10⁶ and Pr = 100. Potential benefits of the TLBM include an ability for higher resolution simulations than can be achieved using classical methods, and faster simulations which may allow phase space studies to determine which parameter combinations lead to which class of behaviour. As the TLBM is a new method for geodynamical simulation, it will take some time to determine the limits of this method. For example, a simulation can be made to run faster by increasing the physical time step, but eventually, if the time step is too large, the Mach numbers on the lattice become too high leading to lower accuracy and eventually instability due to non-convergence of the collision step which involves a relaxation of the number densities to equilibrium. We believe that over time, these limitations will become well understood and that the outstanding parallel scaling performance on HPC CPU clusters of the TLBM - which makes possible 3D models up to 5000³ - will  open up exascale computing to geodynamics research and will lead to fundamental advances in geodynamics research. As such, the TLBM may become a valuable tool to advance geodynamics research into the future through large to exascale simulations that may lead to new insights into the dynamics and evolution of the earth and exoplanets from the early lava world stage through to plate tectonics or other regimes.