Difference approximations for the second order wave equation

Difference approximations are derived for the second order wave equation in one and two space dimensions, without first writing it as a first order system. Both the Dirichlet and the Neumann problems are treated for the one-dimensional case. Relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary. General two-dimensional domains are considered for the Dirichlet problem where the domain is embedded in a Cartesian grid and the boundary conditions are approximated by interpolation. A stable conservative scheme is derived where the time step is determined only by the interior discretization formula. Discretization cells cut by the boundary are treated implicitly, but the resulting scheme becomes explicit because the implicit dependence only is pointwise. Numerical examples are provided to verify the stability and accuracy of the proposed method.