Displacement stability of a symmetric semidiscrete wave equation

Let $K_h$ be a symmetric nonnegative matrix and consider $U^{\prime \prime }=(\alpha I-K_h)U$ with fixed real $\alpha$. In any inner product for which $K_h$ is self-adjoint, diagonalization of a matrix givesIndeed each modal coefficient is either a cosine and sine divided by its frequency, a linear function at zero frequency, or a hyperbolic cosine and sine divided by its growth rate. The bounds $|\cos (\omega t)|\le 1$, $|\sin (\omega t)/\omega |\le t$, and $\sinh (\beta t)/\beta \le t{e}^{\beta t}$ prove the estimate. It is uniform in the spectrum of $K_h$ and therefore gives finite-time displacement stability. Uniform velocity estimates need additional control of the initial spatial energy. All-time displacement bounds require a strictly positive lower bound for $K_h-\alpha I$, a different condition.

For $U^{\prime \prime }=(\alpha I-K_h)U$ with symmetric positive definite $K_h$, all-data displacement is bounded for all time on a fixed grid exactly when $\alpha <{\lambda }_{\min }(K_h)$. Diagonalization of a matrix reduces this assertion to oscillators. At equality, an initial velocity in a zero-frequency mode produces linear growth; above it a growing hyperbolic mode is available. A lower bound on ${\lambda }_{\min }(K_h)-\alpha$ uniform in the mesh gives a uniform displacement estimate. This all-time statement is stronger than displacement stability of a symmetric semidiscrete wave equation on fixed finite time intervals.