Numerical Analysis and Scientific Computing Seminar | Tristan Montoya, An entropy-stable discontinuous spectral-element method for the spherical shallow water equations in covariant form

In this seminar, we introduce a new discontinuous spectral-element formulation which guarantees conservation of mass as well as conservation or dissipation of total energy (i.e. a mathematical entropy function) for the shallow water equations in covariant form, applicable to general unstructured quadrilateral grids on two-dimensional manifolds such as the sphere. We will begin with an introduction to the formulation of systems of balance laws on curved manifolds using concepts from differential geometry. Several approaches to their numerical solution will be reviewed, motivating the development of the proposed methods with a particular view towards applications in geophysical fluid dynamics. We will then derive a novel flux-differencing formulation based on a skew-symmetric splitting of the covariant derivative, which yields an entropy-conservative or entropy-stable discretization of the spherical shallow water equations when used with tensor-product summation-by-parts operators on the reference element. Numerical experiments will be presented in which we verify the discrete conservation and stability properties of the resulting schemes and assess their accuracy and robustness in a series of idealized global atmospheric test cases. We will conclude with an outlook on applications of the proposed methodology to dynamical core development for next-generation weather and climate models.

This talk is based on joint work with Gregor Gassner and Andrés Rueda-Ramírez.