# On the Quality of Velocity Interpolation Schemes for Marker-in-Cell Method and Staggered Grids

## Abstract

The marker-in-cell method is generally considered a flexible and robust method to model the advection of heterogenous non-diffusive properties (i.e., rock type or composition) in geodynamic problems. In this method, Lagrangian points carrying compositional information are advected with the ambient velocity field on an Eulerian grid. However, velocity interpolation from grid points to marker locations is often performed without considering the divergence of the velocity field at the interpolated locations (i.e., non-conservative). Such interpolation schemes can induce non-physical clustering of markers when strong velocity gradients are present (Journal of Computational Physics 166:218–252, 2001) and this may, eventually, result in empty grid cells, a serious numerical violation of the marker-in-cell method. To remedy this at low computational costs, Jenny et al. (Journal of Computational Physics 166:218–252, 2001) and Meyer and Jenny (Proceedings in Applied Mathematics and Mechanics 4:466–467, 2004) proposed a simple, conservative velocity interpolation scheme for 2-D staggered grid, while Wang et al. (Geochemistry, Geophysics, Geosystems 16(6):2015–2023, 2015) extended the formulation to 3-D finite element methods. Here, we adapt this formulation for 3-D staggered grids (correction interpolation) and we report on the quality of various velocity interpolation methods for 2-D and 3-D staggered grids. We test the interpolation schemes in combination with different advection schemes on incompressible Stokes problems with strong velocity gradients, which are discretized using a finite difference method. Our results suggest that a conservative formulation reduces the dispersion and clustering of markers, minimizing the need of unphysical marker control in geodynamic models.

## Keywords

Interpolation Method Interpolation Scheme Stagger Grid Advection Scheme Subduction Channel## Notes

### Acknowledgements

We thank Taras Gerya for sharing the **LinP** interpolation method and for the many discussions related to this paper. We also thank the reviewers Thibault Duretz, Cedric Thieulot, Hongliang Wang, and the editor Wim Spakman, whose comments helped improve this paper. Funding was provided by the European Research Council under the European Community’s Seventh Framework Program (FP7/2007-2013)/ERC Grant Agreement #258830. The simulation data and the codes to reproduce the results presented in this study can be provided on request.

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