Galerkin finite element methods for the Shallow Water equations over variable bottom

Abstract

We consider the one-dimensional shallow water equations (SW) in a finite channel with variable bottom topography. We pose several initialboundary-value problems for the SW system, including problems with transparent (characteristic) boundary conditions in the supercritical and the subcritical case. We discretize these problems in the spatial variable by standard Galerkin-finite element methods and prove L 2 -error estimates for the resulting semidiscrete approximations. We couple the schemes with the 4 th order-accurate, explicit, classical Runge-Kutta time stepping procedure and use the resulting fully discrete methods in numerical experiments of shallow water wave propagation over variable bottom topographies with several kinds of boundary conditions. We discuss issues related to the attainment of a steady state of the simulated flows, including the good balance of the schemes.

Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows

“…where d 0 is the initial height if the parabolic mound, and R 0 is the initial radius of the mound. At time T the central height has reduced to 1 2 d 0 , that follows from the following relation…”

“…Because analytical solutions of the SWE are only available in very few simple and ideal situations, it is important to develop robust and accurate numerical methods to solve SWE in more realistic engineering applications. Extensive research on numerical models for SWE has been developed for a long time, and can generally be classified into three types, i.e., the finite element method [1], the finite difference method [2], or the finite volume method [3].…”

Embedded domain Reduced Basis Models for the shallow water hyperbolic equations with the Shifted Boundary Method

“…Strong formulation of the shallow water problem. The shallow-water equations, also known as de Saint-Venant equations [4], are a system of hyperbolic partial differential equations simulating the behavior of a free surface of a fluid [28,45] when the depth of the fluid bed is shallow compared to the characteristic horizontal spatial length. Such system of equations is derived from the Navier-Stokes equations after integrating through the depth and observing that, since the horizontal length scale is much greater than the vertical length scale, the vertical component of the fluid velocity field is small compared to the horizontal component and the vertical gradients of pressure are nearly hydrostatic.…”

A novel technique for implementing the finite element method in a shallow water equation

“…Therefore, it is crucial to establish reliable, efficient, and practical numerical methods for real-world applications. Extensive study on numerical models for SWE has been conducted, which can be broadly classified into three categories, i.e., the finite element technique [ 12 ], the finite difference method [ 13 ], and the finite volume approximation [ 14 ].…”