12,507 research outputs found
Equilibrium solutions of the shallow water equations
A statistical method for calculating equilibrium solutions of the shallow
water equations, a model of essentially 2-d fluid flow with a free surface, is
described. The model contains a competing acoustic turbulent {\it direct}
energy cascade, and a 2-d turbulent {\it inverse} energy cascade. It is shown,
nonetheless that, just as in the corresponding theory of the inviscid Euler
equation, the infinite number of conserved quantities constrain the flow
sufficiently to produce nontrivial large-scale vortex structures which are
solutions to a set of explicitly derived coupled nonlinear partial differential
equations.Comment: 4 pages, no figures. Submitted to Physical Review Letter
SWASHES: a compilation of Shallow Water Analytic Solutions for Hydraulic and Environmental Studies
Numerous codes are being developed to solve Shallow Water equations. Because
there are used in hydraulic and environmental studies, their capability to
simulate properly flow dynamics is critical to guarantee infrastructure and
human safety. While validating these codes is an important issue, code
validations are currently restricted because analytic solutions to the Shallow
Water equations are rare and have been published on an individual basis over a
period of more than five decades. This article aims at making analytic
solutions to the Shallow Water equations easily available to code developers
and users. It compiles a significant number of analytic solutions to the
Shallow Water equations that are currently scattered through the literature of
various scientific disciplines. The analytic solutions are described in a
unified formalism to make a consistent set of test cases. These analytic
solutions encompass a wide variety of flow conditions (supercritical,
subcritical, shock, etc.), in 1 or 2 space dimensions, with or without rain and
soil friction, for transitory flow or steady state. The corresponding source
codes are made available to the community
(http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow
Water-based models can easily find an adaptable benchmark library to validate
their numerical methods.Comment: 40 pages There are some errors in the published version. This is a
corrected versio
Well-balanced -adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations
In this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on -adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solution is observed. Obtaining well-balanced numerical schemes for the shallow water equations on fixed meshes is nontrivial and has been a topic of much research. In [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887–1922] we introduced a well-balanced discontinuous Galerkin method using the theory of weak solutions for nonconservative products introduced in [G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548]. In this article we continue this approach and prove well-balancedness of a discontinuous Galerkin method for the shallow water equations on moving meshes. Numerical simulations are then performed to verify the -adaptive method in combination with the space-time discontinuous Galerkin method against analytical solutions and showing its robustness on more complex problems
Mathematical derivation of viscous shallow-water equations with zero surface tension
The purpose of this paper is to derive rigorously the so called viscous
shallow water equations given for instance page 958-959 in [A. Oron, S.H.
Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931?980]. Such a system of
equations is similar to compressible Navier-Stokes equations for a barotropic
fluid with a non-constant viscosity. To do that, we consider a layer of
incompressible and Newtonian fluid which is relatively thin, assuming no
surface tension at the free surface. The motion of the fluid is described by 3d
Navier-Stokes equations with constant viscosity and free surface. We prove that
for a set of suitable initial data (asymptotically close to "shallow water
initial data"), the Cauchy problem for these equations is well-posed, and the
solution converges to the solution of viscous shallow water equations. More
precisely, we build the solution of the full problem as a perturbation of the
strong solution to the viscous shallow water equations. The method of proof is
based on a Lagrangian change of variable that fixes the fluid domain and we
have to prove the well-posedness in thin domains: we have to pay a special
attention to constants in classical Sobolev inequalities and regularity in
Stokes problem
An Integrable Model For Undular Bores On Shallow Water
On the basis of the integrable Kaup-Boussinesq version of the shallow water equations, an analytical theory of undular bores is constructed. The problem of the decay of an initial discontinuity is considered
Non-Linear Shallow Water Equations numerical integration on curvilinear boundary-conforming grids
An Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the Shallow Water Equations on generalized curvilinear coordinate systems is proposed. The Shallow Water Equations are expressed in a contravariant formulation in which Christoffel symbols are avoided. The equations are solved by using a high-resolution finite-volume method incorporated with an exact Riemann Solver. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities on generalized boundary-conforming grids is presented; this procedure allows the numerical scheme to satisfy the freestream preservation property on highly-distorted grids. The capacity of the proposed model is verified against test cases present in literature. The results obtained are compared with analytical solutions and alternative numerical solutions
Modified method of characteristics for the shallow water equations
Flow in open channels is frequently modelled using the shallow water equations (SWEs) with an up-winded scheme often used for the nonlinear terms in the numerical scheme (Delis et al., 2000; Erduran et al., 2002). This paper presents a mathematical model based on the SWEs to compute one dimensional (1-D) open channel flow. Two techniques have been used for the simulation of the flood wave along streams which are initially dry. The first one uses up-winding applied to the convective acceleration term in the SWEs to overcome the problem of numerical instabilities. This is applied to the integration of the shallow water equations within the domain, so the scheme does not require any special treatment, such as artificial viscosity or front tracking technique, to capture steep gradients in the solution. As in all initial value problems, the main difficulty is the boundaries, the conventional method of characteristics (MOC) can be applied in a straight forward way for a lot of cases, but when dealing with a very shallow initial depths followed by a flood wave, it is not possible to overcome the problem of reflections. So a modified method of characteristics (MMOC) is the second technique that has been developed by the authors to obtain a fully transparent downstream boundary and is the main subject of this paper. The mathematical model which integrates the SWEs using a staggered finite difference scheme within the domain and the MMOC near the boundary has been tested not only by comparing its results with some analytical solutions for both steady and unsteady flow but also by comparing the results obtained with the results of other models such as Abiola et al. (1988)
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