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

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.