The numerical solution of the wave equation at the first time step

Superimposing a uniformgrid on the space variable in the simple wave equation, for which boundary conditions, initial displacement and initial velocity are specified, leads to the numerical solution of a system of second order ordinary differential equations. Before the numerical solution of this system can proceed, the displacement at the first time level must be estimated from the initial conditions. The accuracy of this starting value must be at least that of the method to be used at higher time levels. In the present note, approximations to the displacement with second, fourth and sixth order accuracy are determined. The procedure followed can be used to obtain yet higher accuracy by choosing other approximations to the initial velocity. Key Word