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.