Geodesic
Stability of numerical methods for solving second-order hyperbolic equations with a small parameter
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 490 (2020), pp. 35-41

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We study a symmetric three-level (in time) method with a weight and a symmetric vector two-level method for solving the initial-boundary value problem for a second-order hyperbolic equation with a small parameter $\tau>0$ multiplying the highest time derivative, where the hyperbolic equation is a perturbation of the corresponding parabolic equation. It is proved that the solutions are uniformly stable in $\tau$ and time in two norms with respect to the initial data and the right-hand side of the equation. Additionally, the case where $\tau$ also multiplies the elliptic part of the equation is covered. The spacial discretization can be performed using the finite-difference or finite element method.
Keywords: second-order hyperbolic equations, small parameter, three- and two-level methods, uniform stability in small parameter and time. 
@article{DANMA_2020_490_a7,
     author = {A. A. Zlotnik and B. N. Chetverushkin},
     title = {Stability of numerical methods for solving second-order hyperbolic equations with a small parameter},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {35--41},
     publisher = {mathdoc},
     volume = {490},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_490_a7/}
}
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A. A. Zlotnik; B. N. Chetverushkin. Stability of numerical methods for solving second-order hyperbolic equations with a small parameter. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 490 (2020), pp. 35-41. http://geodesic.mathdoc.fr/item/DANMA_2020_490_a7/