Error estimates for finite difference approximations to hyperbolic equations for large time

Abstract:

Convergence results for bounded time intervals are well known for finite difference approximations to the Cauchy problem for hyperbolic equations. These results typically state that if the initial data is smooth and the approximation is stable in ${L^2}$ and accurate of order $r$, then the error at time $t$ is bounded by $C(t,f){h^r}$, where $f$ is the initial data and $C(t) = O(t)$. This paper considers the error for long times. It is not possible for the error to be $O({h^r})$ in ${L^p}$ uniformly in $t$. However, it is shown here that if $\Omega$ is a bounded domain the error in ${L^p}(\Omega )$ is bounded by $C(\Omega ,f){h^r}$, where $C$ is independent of $t$. Thus, the global error will grow as more timesteps are taken but the local error will remain uniformly bounded.