A Fourth-Order Compact Implicit Scheme for Solving the Non-Linear Shallow-Water Equations in Conservation-Law Form

1980 ◽  
pp. 211-220
Author(s):  
I. M. Navon
Keyword(s):  
Shallow Water ◽  
Fourth Order ◽  
Shallow Water Equations ◽  
Conservation Law ◽  
Implicit Scheme ◽  
Non Linear

scholarly journals On the standard Galerkin method with explicit RK4 time stepping for the shallow water equations

2019 ◽  
Vol 40 (4) ◽  
pp. 2415-2449
Author(s):  
D C Antonopoulos ◽  
V A Dougalis ◽  
G Kounadis
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Fourth Order ◽  
Shallow Water Equations ◽  
Computational Study ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Courant Number ◽  
Standard Galerkin Method ◽  
Initial Boundary Value

Abstract We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension. We discretize the problem in space by the standard Galerkin finite element method on a quasiuniform mesh and in time by the classical four-stage, fourth order, explicit Runge–Kutta scheme. Assuming smoothness of solutions, a Courant number restriction and certain hypotheses on the finite element spaces, we prove $L^{2}$ error estimates that are of fourth-order accuracy in the temporal variable and of the usual, due to the nonuniform mesh, suboptimal order in space. We also make a computational study of the numerical spatial and temporal orders of convergence, and of the validity of a hypothesis made on the finite element spaces.

scholarly journals A FINITE ELEMENT METHOD FOR STORM SURGE AND TIDAL COMPUTATION

1984 ◽  
Vol 1 (19) ◽  
pp. 82 ◽  
Author(s):  
Y. Coeffe ◽  
S. Dal Secco ◽  
P. Esposito ◽  
B. Latteux
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Shallow Water ◽  
Storm Surge ◽  
Incident Wave ◽  
Shallow Water Equations ◽  
Implicit Scheme ◽  
Wave Condition ◽  
Recent Developments ◽  
Element Method

The paper reports the current progress in developing a finite element method for the shallow water equations. Some recent developments as the implementation of a semi implicit scheme or the use of an incident wave condition are described. Different realistic applications are presented concerning tidal and storm surge simulations.

Non-linear waves interactions in rotating shallow water magnetohydrodynamics

2020 ◽  
Author(s):  
Dmitry Klimachkov ◽  
Arakel Petrosyan
Keyword(s):  
Magnetic Field ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Plasma Layer ◽  
Vertical Component ◽  
Vertical Magnetic Field ◽  
Amplitude Equations ◽  
Horizontal Magnetic Field ◽  
Non Linear ◽  
Poincare Waves

<p>This study is devoted to the development of the nonlinear theory of the magneto-Poincare waves and magnetostrophic waves in rotating layers of astrophysical and space plasma in the shallow-water approximation. These waves determine the large-scale dynamics of the various astrophysical and space objects such as solar tachocline, as well as  magnetoactive atmospheres of exoplanets trapped by tides of a carrier star, neutron stars atmospheres and the flows in accretion disks of neutron stars. For this purpose we derived magnetohydrodynamic shallow water equations with a rotation and presence of an external vertical magnetic field. The system is obtained from conventional magnetohydrodynamic equations for incompressible inviscid heavy plasma layer with free surface in an external vertical magnetic field. The pressure is assumed to be hydrostatic, and the height of the plasma layer is considered to be much smaller than horizontal scales of the flow. The magnetohydrodynamic equations in the shallow-water approximation play equally important role in the space and astrophysical plasma flows like classical shallow-water equations in the fluid dynamics of a neutral fluid. The magnetohydrodynamic shallow water equations with an external vertical magnetic field are modified by supplementing them with the equation for the vertical component of the magnetic field and divergence-free condition for magnetic field contains its vertical component. Thus the velocity field remains two-dimensional while the magnetic field becomes three-dimensional. It is shown that the presence of a vertical magnetic field significantly changes the dynamics of the wave processes in astrophysical plasma compared to the neutral fluid and plasma layer in a horizontal magnetic field.  We have investigated the interaction of Magneto-Poincare waves and magnetostrophic waves in the magnetohydrodynamic shallow water flows in external vertical magnetic field and in horizontal (toroidal and poloidal) magnetic field. In the absence of the horizontal magnetic field the dynamics of plasma appears to be similar to the neutral fluid dynamics and it is shown that there are four-waves interactions in this case. Using the asymptotic multiscale method we obtained the non-linear amplitude equations for the three interacting Magneto-Poincare waves and magnetostrophic waves. The analysis of the amplitude equations shows that there are two types of instabilities for four different types of three-waves configurations. These instabilities occur in both cases: in the external vertical magnetic field and in the horizontal magnetic field. For all types of instabilities the growth rates are found. In the absence of the vertical magnetic field we obtained the non-linear amplitude equations for the four interacting waves. It is shown that the resulting system of equations has the first integrals that describe the mechanism of energy transfer among interacting waves of small amplitude. This work was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).</p>

scholarly journals OSCILLATIONS OF SEMI-ENCLOSED WATER BODY INDUCED BY HURRICANES

2011 ◽  
Vol 1 (32) ◽  
pp. 41 ◽  
Author(s):  
Paul Yuan-Hung Tan ◽  
Jiin-Jen Lee
Keyword(s):  
Shallow Water ◽  
Hurricane Katrina ◽  
Water Body ◽  
Shallow Water Equations ◽  
Present Model ◽  
Storm Surges ◽  
Non Linear ◽  
Major Application ◽  
Primary Focus ◽  
Volume Method

The primary focus of this research is to study the oscillations of semi-enclosed water body induced by hurricanes. The physical mechanisms of the wind-induced oscillation (storm surge) in a semi-enclosed water body are analyzed by the depth-averaged, non-linear shallow-water equations. The numerical model using the finite-volume method (FVM) to solve the depth-averaged, non-linear shallow-water equations is developed and the present model is verified by the intensive field data reported by IPET. The present model is further applied to the investigation of the oscillations (storm surges) in Lake Pontchartrain induced by the winds generated by four synthetic hurricanes within the time-span of 00:00 UTC August 29 to 00:00 UTC August 30, 2005: 1.Hurricane Katrina tracking on its original route, 2.Hurricane Katrina tracking 36 km west of its original route, 3.Hurricane Katrina tracking 72 km west of its original route, and 4.Hurricane Katrina tracking on its original route with reduced forward speeds. The major application of the present model is to assist the design of the water-front structure surrounding the semi-enclosed water body that has been influenced by the oscillations induced by hurricanes. The numerical simulations generated by the present model can help the planners to determine a better strategy of the hurricane protection systems surrounding the communities of the semi-enclosed water body.

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