12/20/2017

# Discrete Conservation Properties for Shallow Water Flows using Mixed Mimetic Spectral Elements

New methods for simulating fluid dynamics in the atmosphere and ocean.

## The Science

This paper explores a new method for the solution of the rotating shallow water equations – a simpler analog often used to test approaches for full general circulation models. This new method exactly preserves conservation properties of the fluid flow with arbitrarily high accuracy and is a strong candidate for use in climate system simulations at long timescales.

## The Impact

The conservation and error convergence properties of this new approach make it a strong candidate for modeling geophysical dynamics over long-time integrations with high accuracy and minimal bias. In particular, it should be well suited to the modeling of both gravity waves on fast time scales and Rossby waves on slow time scales that are responsible for many aspects of climate and weather dynamics.

## Summary

To simulate fluid flow on a computer, we must translate the continuous fluid equations into a discrete form that can be calculated on a computational mesh covering the sphere. Mimetic methods are a technical term for a discrete formulation that can exactly capture properties of the continuous mathematical equations. Spectral element methods are a choice of discrete model that uses a series of carefully chosen mathematical functions, called basis functions, to provide a locally accurate approximation to the solution in each mesh cell (element). In this work, mimetic and spectral element approaches are combined and applied to a set of simpler fluid equations known as a shallow water model. The combined mixed mimetic spectral element method introduces a set of recently developed basis functions for the spectral element approach that exactly satisfy properties of the underlying mathematical operators (i.e., mimetic). This allows for the exact conservation of both simpler quantities (e.g., mass) as well as higher moments like energy or enstrophy, subject to the truncation error of the time stepping scheme. The combination of very high accuracy associated with spectral element methods and the preservation of conservation properties needed to represent the Earth system in long timescale simulations appears to provide a much improved method for solving the fluid equations for use in large-scale Earth system models. This work has provided an initial implementation of these idea and has demonstrated much improved results. Future work will integrate these schemes in the full dynamical cores of Earth system models.

## Principal Investigator(s)

Phil Jones

Los Alamos National Laboratory

pwjones@lanl.gov

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## Funding

Funding was provided by the Earth System Modeling program within the DOE Office of Science Office of Biological and Environmental Research.

## References

Lee, D., A. Palha, and M. Gerritsma. “Discrete conservation properties for shallow water flows using mixed mimetic spectral elements.” *Journal of Computational Physics* **357**, 282-304 (2017). DOI:10.1016/j.jcp.2017.12.022