Geodesic
CABARET scheme with improved dispersion properties for systems of linear hyperbolic-type differential equations
Numerical methods and programming, Tome 22 (2021) no. 1, pp. 67-76.

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A conservative-characteristic method to solve systems of linear hyperbolic-type partial differential equations is proposed. This method has the fourth order of approximation on uniform grids and the second approximation order and improved dispersion properties on non-uniform grids. The proposed method is based on the well-known CABARET scheme whose conservative phases are modified by adding anti-dispersive terms of a special type. Previously, a method with similar properties was proposed only for the simplest one-dimensional linear advection equation. The modification of the scheme allows us to improve the dispersion properties of the advection for all Riemann invariants of the system of equations under consideration at once. The scheme is non-dissipative when the monotonization procedures are not used and is stable at Courant numbers $CFL \le 1$. The accuracy of the method and its order of convergence are shown in a series of solving the problem of advection of a wave modulated by a Gaussian on a sequence of condensing grids. The proposed method is planned to be used as a basis for constructing a CABARET scheme with improved dispersion properties for systems of nonlinear differential equations.
Keywords: conservative-characteristic methods; CABARET scheme; Computational Fluid Dynamics (CFD); high-order schemes; hyperbolic-type equations. 
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     author = {N. A. Afanasiev and V. M. Goloviznin and A. V. Solov'ev},
     title = {CABARET scheme with improved dispersion properties for systems of linear hyperbolic-type differential equations},
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     pages = {67--76},
     publisher = {mathdoc},
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     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2021_22_1_a2/}
}
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N. A. Afanasiev; V. M. Goloviznin; A. V. Solov'ev. CABARET scheme with improved dispersion properties for systems of linear hyperbolic-type differential equations. Numerical methods and programming, Tome 22 (2021) no. 1, pp. 67-76. http://geodesic.mathdoc.fr/item/VMP_2021_22_1_a2/