A space–time CESE scheme for shallow water magnetohydrodynamics equations with variable bottom topography

In this paper, a mathematical model of stratified geophysical fluid flow over variable bottom topography was derived for shallow water. The equations are derived from the principles of conservation of mass and conservation of momentum. The force acting on the fluid is gravity, represented by the gravitational constant. A system of six nonlinear partial differential equations was obtained as the model equations. The solutions of these models were obtained using perturbation method. The presence of the coriolis force in the shallow water equations were shown as the causes of the deflection of fluid parcels in the direction of wave motion and causes gravity waves to disperse. As water depth decreases due to varied bottom topography, the wave amplitude were shown to increase while the wavelength and wave speed decreases resulting in overturning of the wave. The results are presented graphically.

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Zeroth‐order conservation laws of two‐dimensional shallow water equations with variable bottom topography

Studies in Applied Mathematics ◽

10.1111/sapm.12320 ◽

2020 ◽

Vol 145 (2) ◽

pp. 291-321

Author(s):

Alexander Bihlo ◽

Roman O. Popovych

Keyword(s):

Conservation Laws ◽

Shallow Water ◽

Shallow Water Equations ◽

Bottom Topography ◽

Two Dimensional ◽

Zeroth Order ◽

Variable Bottom

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ADVANCED HYPERBOLIC DIVERGENCE CLEANING SCHEME FOR SHALLOW WATER MAGNETOHYDRODYNAMICS

Journal of Hyperbolic Differential Equations ◽

10.1142/s021989160400007x ◽

2004 ◽

Vol 01 (01) ◽

pp. 171-195 ◽

Cited By ~ 2

Author(s):

YONG-JOONG LEE ◽

CLAUS-DIETER MUNZ

Keyword(s):

Shallow Water ◽

Lagrange Multiplier ◽

Operator Splitting ◽

Computational Time ◽

Computational Domain ◽

Step Method ◽

Time Step ◽

Magnetohydrodynamics Equations ◽

Shallow Water Magnetohydrodynamics ◽

Shock Dissipation

An advanced hyperbolic divergence cleaning scheme based on "generalized Lagrange multiplier" (GLM) for the equations of "shallow water magnetohydrodynamics" (SMHD) is presented. This scheme is based on the two-step method which is comprised of the standard finite-volume updating step for the nonlinear genuine SMHD system and the divergence cleaning step for the linear GLM-based Maxwell subsystem. The divergence cleaning step can be applied several times per each computational time step, in order to accelerate the transports of the divergence error out of the computational domain. The presented two-step method is compared with the standard GLM method based on operator splitting. It is shown that the standard operator-splitting based method has the shock dissipation problem, particularly when the multiple subcycles of the divergence cleaning step is performed per each time step. On the contrary, the introduction of the multiple subcycles for the new GLM–Maxwell subsystem does not suffer from the dissipation of the shocks, and produces better shock resolution. The presented method can be further applied to the full magnetohydrodynamics equations.

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