New high order time-stepping schemes for finite differences

To improve finite difference calculations, the use of higher order difference operators is promising. For the time-stepping, usually only the second order centred differencing is used. Since this leads only to second order accurate schemes, independent of the choice of the difference operator, we investigate classical explicit higher order multistep methods. Unfortunately, they do not satisfy completely the needs for the use with Maxwell's or telegraph equations. Therefore, we construct new explicit multistep schemes, which show stability limits comparable to centred differencing and are more adapted to our purposes. This paper discusses the construction principles of the new schemes, gives the stability limits as well as the convergence orders and error constants, and validates the approach by applying the schemes to telegraph equations using several difference operators for the approximation of the spatial derivatives.