# CAIMS Prize Award

Institute of Applied Mathematics, University of British Columbia

Supervisor: Wei-Hua Hsieh

Thesis title: Analysis of a Galerkin-Characteristic Algorithm for the Potential Vorticity - Stream Function Equations

Abstract:

In this thesis we develop and analyse 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 yields 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 feet of the characteristic curves of the advective component of the equations. The algorithm is unconditionally stable and of order O(h^{4}/k), where k is the size of the time step and h is the size of the space discretization parameter. The second stage of the algorithm is a projection of the Lagrangian representation of the flow onto the Cartesian space-time Eulerian representation coordinated with Crank-Nicolson Finite Elements. The error analysis for this stage in the L^{2}-norm shows that the approximation component of the global error is O(h^{2}) 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 C(k^{2}+h), where C is a constant that depends on the derivatives of the advected quantity along the characteristics, so that C is in general small. Numerical experiments illustrate our theoretical results for both stages of the method.