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Abstract:
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In recent years, there has been much interest in the numerical solution of shallow water equations. The numerical procedure used to solve the shallow water equations must resolve the physics of the problem without introducing spurious oscillations or excessive numerical diffusion. Staggered-grid finite difference methods have been used extensively in modeling surface flow without introducing spurious oscillations. Finite element methods, permitting a high degree of grid flexibility for complex geometries and facilitating grid refinement near land boundaries to resolve important processes, have become much more prevalent. However, early finite element simulations of shallow water systems were plagued by spurious oscillations and the various methods introduced to eliminate these oscillations through artificial diffusion were generally unsuccessful due to excessive damping of physical components of the solution.
Here, we give a brief overview on some finite element models of the shallow water equations, with particular attention given to the wave and characteristic formulations. In the literature, standard analysis, based on Fourier decompositions of these methods, has always neglected contributions from the nonlinear terms.
We derive ${\cal L}\sp{\infty} ((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for both the continuous-time and discrete-time Galerkin approximation to the nonlinear wave model, finding these to be optimal in ${\cal H}\sp1(\Omega).$
Finally, we derive ${\cal L}\sp{\infty}((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for our proposed Characteristic-Galerkin approximation to the nonlinear primitive model. We find these estimates to be optimal in ${\cal H}\sp1(\Omega)$ but with less restrictive time-step constraints when compared to the Galerkin estimates for the wave model. |
