Geodesic
A discrete analog of the Lyapunov function for hyperbolic systems
Contemporary Mathematics. Fundamental Directions, Contemporary problems in mathematics and physics, Tome 64 (2018) no. 4, pp. 591-602

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We study the difference splitting scheme for the numerical calculation of stable solutions of a two-dimensional linear hyperbolic system with dissipative boundary conditions in the case of constant coefficients with lower terms. A discrete analog of the Lyapunov function is constructed and an a priori estimate is obtained for it. The obtained a priori estimate allows us to assert the exponential stability of the numerical solution. 
@article{CMFD_2018_64_4_a0,
     author = {R. D. Aloev and M. U. Khudayberganov},
     title = {A discrete analog of the {Lyapunov} function for hyperbolic systems},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {591--602},
     publisher = {mathdoc},
     volume = {64},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2018_64_4_a0/}
}
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R. D. Aloev; M. U. Khudayberganov. A discrete analog of the Lyapunov function for hyperbolic systems. Contemporary Mathematics. Fundamental Directions, Contemporary problems in mathematics and physics, Tome 64 (2018) no. 4, pp. 591-602. http://geodesic.mathdoc.fr/item/CMFD_2018_64_4_a0/