Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

G. Kounadis, V. A. Dougalis

TL;DR

The paper develops and analyzes standard Galerkin finite element discretizations for the 1D shallow water equations over variable bottom topography, on a finite interval with multiple boundary setups including transparent (characteristic) conditions for both supercritical and subcritical flows. It provides rigorous $L^2$-error estimates for semidiscrete schemes and demonstrates stable, high-order accurate fully discrete methods by coupling with the classical RK4 time stepping, complemented by numerical experiments that show correct wave propagation, steady-state attainment, and good-balance behavior. The subcritical case is treated via diagonalization into variables $v,w$, with corresponding convergence results. The work offers guidance on boundary treatment and quadrature requirements necessary to preserve steady states and balance in simulations of shallow-water waves over varying bottom topography, contributing a robust FE framework for long-time integration and analysis.

Abstract

We consider the one-dimensional shallow water equations (SW) in a finite channel with variable bottom topography. We pose several initial-boundary-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 4th 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.

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

TL;DR

The paper develops and analyzes standard Galerkin finite element discretizations for the 1D shallow water equations over variable bottom topography, on a finite interval with multiple boundary setups including transparent (characteristic) conditions for both supercritical and subcritical flows. It provides rigorous -error estimates for semidiscrete schemes and demonstrates stable, high-order accurate fully discrete methods by coupling with the classical RK4 time stepping, complemented by numerical experiments that show correct wave propagation, steady-state attainment, and good-balance behavior. The subcritical case is treated via diagonalization into variables , with corresponding convergence results. The work offers guidance on boundary treatment and quadrature requirements necessary to preserve steady states and balance in simulations of shallow-water waves over varying bottom topography, contributing a robust FE framework for long-time integration and analysis.

Abstract

We consider the one-dimensional shallow water equations (SW) in a finite channel with variable bottom topography. We pose several initial-boundary-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 4th 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.

Paper Structure

This paper contains 8 sections, 3 theorems, 70 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Initial-boundary-value problems and error estimates
  3. Semidiscretization of a simple ibvp with vanishing fluid velocity at the endpoints
  4. Semidiscretization of an ibvp with absorbing (characteristic) boundary conditions in the supercritical case
  5. Semidiscretization in the case of absorbing (characteristic) boundary conditions in the subcritical case
  6. Numerical experiments
  7. Absorbing (characteristic) boundary conditions
  8. Shallow water equations in balance-law form

Key Result

Proposition 2.1

Let $(\eta,u)$ be the solution of eq:2p1, assumed to be sufficiently smooth and satisfying $\beta+\eta>0$ in $[0,1]\times[0,T]$, where $\beta\in C^1$, $\beta>0$. Let $r\geq 3$ and $h$ be sufficiently small. Then, the semidiscrete ivp eq:2p5--eq:2p6 has a unique solution $(\eta_h,u_h)$ for $t\in [0,T where, here and in the sequel, $C$ will denote a generic constant independent of $h$.

Figures (6)

  • Figure 1: Evolution with data \ref{['eq:3p3']}, supercritical case, $r=2$, $h=1/400$, $k=h/3$
  • Figure 2: Evolution with data \ref{['eq:3p6']}, supercritical case, $r=2$, $h=1/400$, $k=h/3$
  • Figure 3: Evolution with data \ref{['eq:3p7']}, subcritical case, $r=2$, $h=1/2000$, $k=h/10$
  • Figure 4: Evolution with data \ref{['eq:3p8']}, subcritical case, $r=2$, $h=1/2000$, $k=h/10$
  • Figure 5: Supercritical flows over a trapezoidal bottom, \ref{['eq:3p9']}--\ref{['eq:3p10']}, $r=2$, $h=1000\,\mathrm m$, $k = 1\,\mathrm s$. (Upper figures: steady-state $\eta(x)$; lower figures: $\beta(x)$.)
  • ...and 1 more figures

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof