Generalized Energy and Potential Enstrophy Conserving Finite Difference Schemes for the Shallow Water Equations

1988 ◽  
Vol 116 (3) ◽  
pp. 650-662 ◽  
Author(s):  
Frank Abramopoulos
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Potential Enstrophy

Finite difference methods for 2D shallow water equations with dispersion

2019 ◽  
Vol 34 (2) ◽  
pp. 105-117 ◽  
Author(s):  
Gayaz S. Khakimzyanov ◽  
Zinaida I. Fedotova ◽  
Oleg I. Gusev ◽  
Nina Yu. Shokina
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Original System ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Elliptic Type ◽  
Two Dimensional ◽  
Nonlinear Hyperbolic System

Abstract Basic properties of some finite difference schemes for two-dimensional nonlinear dispersive equations for hydrodynamics of surface waves are considered. It is shown that stability conditions for difference schemes of shallow water equations are qualitatively different in the cases the dispersion is taken into account, or not. The difference in the behavior of phase errors in one- and two-dimensional cases is pointed out. Special attention is paid to the numerical algorithm based on the splitting of the original system of equations into a nonlinear hyperbolic system and a scalar linear equation of elliptic type.

Characteristics of finite difference methods for dispersive shallow water equations

2016 ◽  
Vol 31 (3) ◽  
Author(s):  
Zinaida I. Fedotova ◽  
Gayaz S. Khakimzyanov
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hydrodynamic Equations ◽  
Difference Methods

AbstractThe paper contains a description of the most important properties of numerical methods for solving nonlinear dispersive hydrodynamic equations and their distinctions from similar properties of finite difference schemes approximating classic dispersion-free shallow water equations.

On the Spectral Convergence of the Supercompact Finite-Difference Schemes for the f-Plane Shallow-Water Equations

2009 ◽  
Vol 137 (7) ◽  
pp. 2393-2406 ◽  
Author(s):  
S. Ghader ◽  
A. R. Mohebalhojeh ◽  
V. Esfahanian
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Small Scale ◽  
Finite Difference Schemes ◽  
Vorticity Field ◽  
Eighth Order ◽  
Spectral Convergence ◽  
Nonlinear Flows

For the f-plane shallow-water equations, the convergence properties of the supercompact finite-difference method (SCFDM) are examined during the evolution of complex, nonlinear flows spawned by an unstable jet. The second-, fourth-, sixth-, and eighth-order SCFDMs are compared with a standard pseudospectral (PS) method. To control the buildup of small-scale activity and thus the potential for numerical instability, the vorticity field is damped explicitly by the application of a triharmonic hyperdiffusion operator acting on the vorticity field. The global distribution of mass between isolevels of potential vorticity, called mass error, and the representation of the balance and imbalance are used to assess numerical accuracy. In each of the quantitative measures, a clear convergence of the SCFDM to the PS method is observed. There is no saturation in accuracy up to the eighth order examined. Taking the PS solution as the reference, for the fundamental quantity of potential vorticity the rate of convergence to PS turns out to be algebraic and near-quadratic.

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