Geodesic
One unconditionally stable CABARET class scheme for shallow water equations
Numerical methods and programming, Tome 25 (2024) no. 1, pp. 64-77.

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The paper considers a new approach to the construction of implicit unconditionally stable schemes within the framework of the CABARET balance-characteristic technique in relation to the system of shallow water equations. The method is based on the idea of inverting coordinate axes in the CABARET scheme to overcome the time step limitation. The system of equations is nonlinear, since the equations include minmax limiting operations based on the maximum principle for local Riemann invariants. This limitation significantly improves the dispersion properties of the numerical scheme. The nonlinear system of equations is solved using the marching order method. The paper presents the derivation of the numerical scheme for the Courant–Friedrichs–Lewy number $CFL \le 1$ and $CFL > 1$. Test calculations are presented on the simplest transport equation and one-dimensional shallow water problems for the subsonic case. Conclusions are drawn about the influence of nonlinear flux correction on the solution.
Keywords: implicit CABARET scheme; balance-characteristic method; transport equation; shallow water equations 
@article{VMP_2024_25_1_a5,
     author = {A. V. Solovjev and D. G. Asfandiyarov},
     title = {One unconditionally stable {CABARET} class scheme for shallow water equations},
     journal = {Numerical methods and programming},
     pages = {64--77},
     publisher = {mathdoc},
     volume = {25},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2024_25_1_a5/}
}
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A. V. Solovjev; D. G. Asfandiyarov. One unconditionally stable CABARET class scheme for shallow water equations. Numerical methods and programming, Tome 25 (2024) no. 1, pp. 64-77. http://geodesic.mathdoc.fr/item/VMP_2024_25_1_a5/