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## Stability of numerical methods for solving second-order hyperbolic equations with a small parameter

Doklady Mathematics. 2020. Vol. 101. No. 1. P. 30-35.
Zlotnik A. A., Chetverushkin B. N.
Translator: I. Ruzanova.

We study the symmetric three-level with a weight and vector two-level in time methods for solving the initial-boundary value problem for a 2nd order hyperbolic equation with a small parameter $\tau>0$ in front of the higher time derivative which is a perturbation of the corresponding parabolic equation. We prove theorems on the uniform in $\tau$ and time stability of solutions in two norms  with respect to the initial data and the right-hand side of the equation. We also cover the case where $\tau$ stands also in front of the elliptic part of the equation. The spacial discretization can be accomplished both by the finite-difference method and by the finite element method.