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. 
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     title = {Stability of numerical methods for solving second-order hyperbolic equations with a small parameter},
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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/

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