Title: The opportunities and limitations of structure preserving variational methods for geophysical simulation
Numerical methods that preserve geometric properties of the equations of motion have the potential to mitigate against biases and improve the longterm statistics of atmospheric models. By preserving the skew-symmetric structure of the non-canonical Hamiltonian form of the equations of motion important dynamical properties such as the balance of energetic exchanges and the orthogonality of rotational and divergent dynamics may be exactly satisfied in space and time. Additionally, spurious numerical artifacts associated with nonlinear hyperbolic dynamics may be suppressed in an energetically consistent manner by selectively up winding materially conserved quantities that manifest within the Poisson bracket so as to dampen additional invariants (ie: potential enstrophy, entropy) without breaking energy conservation.
The key ingredients to the development of such schemes are mimetic spatial discretisations that allow for the preservation of vector calculus identities and balance laws via compatible mappings between Hilbert spaces, and a discrete gradient method which allows for the exact temporal integration of the variational derivatives of the Hamiltonian. In the present work a mixed mimetic spectral element discretisation is employed, in conjunction with a preconditioning strategy and splitting schemes that allow for the efficient solution of semi-implicit non-linear hyperbolic systems. Results are presented for standard test cases for the rotating shallow water equations as well as the 3D compressible Euler equations on the sphere at both planetary and non-hydrostatic scales.
David Lee completed his PhD in applied mathematics at Monash University in 2014, before undertaking post-doctoral positions within the Climate, Ocean and Sea-Ice Modelling group at Los Alamos National Laboratory and the Department of Mechanical Engineering at Monash. He is currently a member of the Model Systems team at the Bureau of Meteorology.