It takes three to tango: 1. Simulating buoyancy-driven flow in the presence of large viscosity contrasts

Buoyancy-driven flow is of fundamental importance for numerous geodynamic phenomena. Since the equations of motion governing multiphase flow are rarely amenable to analytical solutions, numerical simulations provide a compelling alternative. They offer the ability to carefully analyze flow phenomena under differing regimes, initial conditions, and flow dynamics. The three key challenges in these computations are (1) the accurate solution of the equations of motion in the presence of large viscosity contrasts, (2) the representation of strongly deforming interfaces between different fluids, and (3) the accurate coupling of fluid and interface solver. In three dimensions, these challenges become even more intricate, and the appropriate choice of numerical scheme has a profound influence on the tractability, accuracy, robustness, and efficiency of the computational simulation. This is the first paper of two that examine numerical simulations of buoyancy-driven flow in the presence of large viscosity contrasts. In this paper, we present our numerical approach which tackles the above three main challenges through a combination of three numerical methods, namely, (1) an extended ghost fluid type discretization which we developed specifically for the Stokes regime, (2) the level set method, and (3) the extension velocity technique. We find that all three components are crucial to obtain a versatile numerical tool for simulating complex structures in evolving flow. We validate our code by reproducing four benchmark problems in two and three dimensions. We devote special attention to comparing our method to other existing techniques, detailing the advantages of this approach. Finally, we highlight several types of geophysical flow problems for which we believe our method to be well suited.