Daniel Y. Le Roux

Analysis of Numerically Induced Oscillations in 2D Finite-Element Shallow-Water Models Part I: Inertia-Gravity Waves

The numerical approximation of shallow-water models is a delicate problem. For most of the discretization schemes, the coupling between the momentum and the continuity equations usually leads to anomalous dispersion in the representation of fast waves. A dispersion relation analysis is employed here to ascertain the presence and determine the form of spurious modes as well as the dispersive nature of the finite-element Galerkin mixed formulation of the two-dimensional linearized shallow-water equations. Nine popular finite-element pairs are considered using a variety of mixed interpolation schemes. For each pair the frequency or dispersion relation is obtained and analyzed, and the dispersion properties are compared analytically and graphically with the continuous case. It is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of fast waves. The

Analysis of Numerically Induced Oscillations in Two-Dimensional Finite-Element Shallow-Water Models Part II: Free Planetary Waves

P^{NC}_{1} - P^{}_{1}

Kernel Analysis of the Discretized Finite Difference and Finite Element Shallow-Water Models

and RT0 pairs are identified as a promising compromise, provided the grid resolution is high relative to the Rossby radius of deformation for the RT0 element. The numerical solutions of two test problems to simulate fast waves are in good agreement with the analytical results.

A New Triangular Finite-Element with Optimum Constraint Ratio for Compressible Fluids

The behavior of planetary (Rossby) waves in finite-element numerical models is investigated by using a quasi-geostrophic approximation to the midlatitude,

Time Discretization Schemes for Poincaré Waves in Finite-Element Shallow-Water Models

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Dispersion Relation Analysis of the

-plane, two-dimensional linear shallow-water equations. A dispersion relation analysis is employed here to determine the possible occurrence of spurious modes and to ascertain the dispersive/dissipative nature of the finite-element Galerkin mixed formulation. Three finite-element pairs are considered by using a variety of mixed interpolation schemes. For each pair the frequency or dispersion relation is obtained and analyzed, and the dispersion properties are compared analytically and graphically with the continuous case. It is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of slow Rossby waves, due to the coupling between the momentum and continuity equations. Numerical solutions of two test problems to simulate slow Rossby waves are in good agreement with the analytical results.

P^NC_1 - P^_1

A constructive linear algebra approach is developed to characterize the kernels of the discretized shallow-water equations. Three kernel relations are identified as necessary conditions for the discretized system to share the same stationary properties as the continuous system. This matrix kernel scheme is computed using MATLAB and applied to investigate the presence, number, and structure of spurious modes arising in typical finite difference and finite element schemes. The kernel concept is then used to characterize the smallest representable vortices for several representative discrete finite difference and finite element schemes. Both uniform and unstructured mesh situations are considered and compared. Numerical experiments are consistent with the analytic results.

Finite-Element Pair in Shallow-Water Models

The discretization of the shallow-water equations using the finite-element method is a delicate problem. Apart from the possible occurrence of pressure and/or velocity modes, other spurious modes may appear that are essentially a consequence of having more momentum than continuity discretized equations, contrary to the continuum case. In this paper a new triangular finite-element pair is proposed which overcomes this imbalance problem. The new pair is shown to improve on results obtained with existing pairs in representing the propagation of fast gravity and slow Rossby waves by discretizing the linear shallow-water equations.

An ecien t Eulerian nite element method for the shallow water equations

The finite-element spatial discretization of the linear shallow-water equations is examined in the context of several temporal discretization schemes. Three finite-element pairs are considered, namely, the

Stability/dispersion analysis of the discontinuous Galerkin linearized shallow‐water system

P^{}_{0}-P^{}_{1}