A remark on finite volume methods for 2D shallow water equations over irregular bottom topography

Abstract

The 2D shallow water equations are often solved through finite volume (FV) methods in the presence of irregular topography or in non-rectangular channels. The ability of FV schemes to preserve uniform flow conditions under these circumstances is herein analysed as a preliminary condition for more involved applications. The widely used standard Harten-Lax-Van Leer (HLL) and a well-balanced Roe method are considered, along with a scheme from the class of wave propagation methods. The results indicate that, differently from the other solvers, the standard HLL does not preserve the uniform condition. It is therefore suggested to benchmark numerical schemes under this condition to avoid inaccurate hydrodynamic simulations. The analysis of the discretized equations reveals an unbalance of the HLL transversal fluxes in the streamwise momentum equation. Finally, a modification of the HLL scheme, based on the auxiliary variable balance method, is proposed, which strongly improves the scheme performance.

Keywords:

• Auxiliary variables
• finite volume method
• irregular topography
• shallow water equations

This article is referred to by:

Closure to “A remark on finite volume methods for 2D shallow water equations over irregular bottom topography” by Cristiana Di Cristo, Massimo Greco, Michele Iervolino, Riccardo Martino and Andrea Vacca

A remark on finite volume methods for 2D shallow water equations over irregular bottom topography By Cristiana Di Cristo, Massimo Greco, Michele Iervolino, Riccardo Martino and Andrea Vacca

Notation

A	=	cell area (m2)
a	=	weighting factor (–)
C	=	Chezy coefficient (–)
D	=	diffusion matrix (–)
d	=	interface length (m)
e	=	left eigenvector
E	=	vectorial flux function with components F and G along x and y, respectively
F, G	=	flux of conserved variables along x and y, respectively
g	=	gravity acceleration (m s−2)
h	=	flow depth (m)
I	=	identity matrix (–)
L	=	length of the computational domain in the x direction (m)
n	=	unit normal vector (–)
q	=	unit width discharge vector with components q and p along x and y, respectively (m2 s−1)
r	=	right eigenvector
R	=	matrix of right eigenvectors
S	=	source term
Sb	=	bottom slope (–)
t	=	time (s)
u, v	=	velocity components along x and y, respectively (m s−1)
U	=	vector of conserved variables
V	=	vector of auxiliary variables
x, y	=	orthogonal coordinates in the horizontal plane (m)
z	=	bottom elevation (m)
$\dot{\alpha }$	=	coefficient of eigenvectors decomposition
$\mathit{\alpha }$	=	vector of eigenvectors decomposition coefficients
$\text{β}$	=	coefficient of eigenvectors decomposition
$\text{β}$	=	vector of eigenvectors decomposition coefficients
Δt	=	timestep for integration (s)
Δx, Δy	=	spatial resolution along x and y, respectively (m)
η	=	free surface elevation (m)
λ	=	characteristic eigenvalue (m s−1)

Subscripts

aux	=	based on the auxiliary variable method
i, j	=	indexes of the computational cell along x and y, respectively
k	=	wave index
l	=	cell interface index
max	=	maximum in the cross section
min	=	minimum along the streamwise direction
x, y	=	relative to x or y direction, respectively

Superscripts

L, R	=	left and right side of the interface, respectively
n	=	discrete time index
s	=	characteristic value for source term evaluation
T	=	transpose
x, y	=	relative to x or y direction, respectively
*	=	numerical
$-$	=	cell-averaged
∼	=	evaluated at the Roe state
$\stackrel{ˆ}{}$	=	evaluated at the discrete level

Additional information

Funding

This research was carried out in the framework of the project MISALVA, funded by the Ministry of the Environment and Protection of Natural Resources [grant no CUP H36C18000970005].