The maximal accuracy of stable difference schemes for the wave equation (Q1893945)

The following symmetric full discretization \[ \sum^S_{j = -S} b_j U_{n+1, m+j} + \sum^s_{j = -s} a_j U_{n,m + j} + \sum^S_{j = -S} b_j U_{n - 1, m + j} = 0 \] for the wave equation \(u_{tt} = a^2 u_{xx}\) is investigated. It is well known that the maximal order of accuracy \(p\) of such a scheme is given by \(p \leq 2(s + S)\). The authors show that the requirement of stability does not reduce this maximal order for any pair \((s,S)\). Furthermore, Padé schemes are proved to be stable for \(0 < \mu < 1\), where \(\mu\) is the Courant number. These schemes can be implemented with high-order absorbing boundary conditions without reducing the range of \(\mu\).