Stability and accuracy for a numerical method approximating an initial boundary value problem are inherently linked together. Stability means that perturbations have a bounded effect on the discrete solution, and is usually characterized by a precise estimate of norms. Such an estimate can be directly used to quantify the accuracy of the method. A very convenient and common way to investigate stability, and hence accuracy, is to use the energy method. If this approach fails one may instead attempt to get results by Laplace transforming in time and using normal mode analysis. Such analysis is usually more involved, but sharper results may follow. In this talk we will show two examples where, even though the energy method is applicable, it is rewarding to consider the problem in the Laplace domain. In the first case we get sharper accuracy results, and in the second case we get sharper temporal bounds.