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UBC Theses and Dissertations

Analysis of a Galerkin-Characteristic algorithm for the potential vorticity-stream function equations Bermejo, Rodolfo


In this thesis we develop and analyze a Galerkin-Characteristic method to integrate the potential vorticity equations of a baroclinic ocean. The method proposed is a two stage inductive algorithm. In the first stage the material derivative of the potential vorticity is approximated by combining Galerkin-Characteristic and Particle methods. This yield a computationally efficient algorithm for this stage. Such an algorithm consists of updating the dependent variable at the grid points by cubic spline interpolation at the foot of the characteristic curves of the advective component of the equations. The algorithm is unconditionally stable and conservative for Δt = O(h). The error analysis with respect to L² -norm shows that the algorithm converges with order O(h); however, in the maximum norm it is proved that for sufficiently smooth functions the foot of the characteristic curves are superconvergent points of order O(h⁴ /Δt). The second stage of the algorithm is a projection of the Lagrangian representation of the flow onto the Cartesian space-time Eularian representation coordinated with Crank-Nicholson Finite Elements. The error analysis for this stage with respect to L²-norm shows that the approximation component of the global error is O(h²) for the free-slip boundary condition, and O(h) for the no-slip boundary condition. These estimates represent an improvement with respect to other estimates for the vorticity previously reported in the literature. The evolutionary component of the global error is equal to K(Δt² + h), where K is a constant that depends on the derivatives of the advective quantity along the Characteristic. Since the potential vorticity is a quasi-conservative quantitiy, one can conclude that K is in general small. Numerical experiments illustrate our theoretical results for both stages of the method.

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