Stability Analysis of an Upwind Difference Splitting Scheme for Two-Dimensional Saint–Venant Equations

The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modelin...

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Bibliographic Details
Published in:Symmetry (Basel) 2022-10, Vol.14 (10), p.1986
Main Authors: Berdyshev, Abdumauvlen, Aloev, Rakhmatillo, Bliyeva, Dana, Dadabayev, Sardor, Baishemirov, Zharasbek
Format: Article
Language:English
Subjects:
Boundary conditions
Boundary value problems
Dependent variables
Flow velocity
Hydraulic engineering
Hydrologic cycle
Mathematical models
Matrices (mathematics)
Methods
Norms
Numerical methods
Ocean models
Oceans
Saint–Venant equations
splitting scheme
stability
Stability analysis
upwind difference scheme
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Summary:The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym14101986