Bachini, Elena (2019) Numerical methods for Shallow Water Equations on regular surfaces. [Ph.D. thesis]
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Abstract (italian or english)
Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this thesis we derive an intrinsic shallow water model starting from the Navier-Stokes equations defined on a local reference frame anchored on the bottom surface. The resulting equations are characterized by non-autonomous flux functions and source terms embodying only geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier-Stokes equations with respect to a geometry-based order parameter.
We then derive numerical discretization schemes compatibles with the intrinsic setting of the formulation, starting from studying a first order upwind Godunov Finite Volume scheme intrinsically defined on the bottom surface. We analyze convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures.
The low-order discretization method is subsequently extended to the Discontinuous Galerkin framework. We implement a linear version of the DG scheme defined intrinsically on the surface and we start from the resolution of the scalar transport equation. We test the scheme for convergence and then we move towards the intrinsic shallow water model. Simulations on synthetic test cases are reported and the improvement with respect to the first order finite volume discretization is clearly visible.
Finally, we consider a finite element method for advection-diffusion-reaction equations on surfaces. Unlike many previous techniques, this approach is based on the geometrically intrinsic formulation and the resulting finite element method is fully intrinsic to the surface. In the last part of this work, we lay out in detail the formulation and compare it to a well-established finite element scheme for surface PDEs. We then evaluate the method for several steady and transient problems involving both diffusion and advection-dominated regime. The experimental results show the theoretically expected convergence rates and good performance of the established finite element methods.
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