Advantages of a Conservative Velocity Interpolation (CVI) Scheme for Particle-in-cell Methods with Application in Geodynamic Modeling

The particle-in-cell method is generally considered a flexible and robust method to model the geodynamic problems with chemical heterogeneity. However, velocity interpolation from grid points to particle locations is often performed without considering the divergence of the velocity field, which can lead to significant particle dispersion or clustering if those particles move through regions of strong velocity gradients. This may ultimately result in cells void of particles, which, if left untreated, may, in turn, lead to numerical inaccuracies. Here we apply a two-dimensional conservative velocity interpolation (CVI) scheme to steady state and time-dependent flow fields with strong velocity gradients (e.g., due to large local viscosity variation) and derive and apply the three-dimensional equivalent. We show that the introduction of CVI significantly reduces the dispersion and clustering of particles in both steady state and time-dependent flow problems and maintains a locally steady number of particles, without the need for ad hoc remedies such as very high initial particle densities or reseeding during the calculation. We illustrate that this method provides a significant improvement to particle distributions in common geodynamic modeling problems such as subduction zones or lithosphere-asthenosphere boundary dynamics.