Jacobi iteration in implicit difference schemes for the wave equation

It is shown how a Jacobi iteration can be used to approximate the linear algebra required by a class of implicit finite difference/element schemes for a linear, hyperbolic PDE while receiving the important properties of the implicit scheme. In particular, it is shown that order of accuracy, stability, energy conservation, and isotropy of the implicit scheme are all carried over into the new scheme. In general, the new scheme is much more efficient than the implicit scheme it is arrived from for problems in more than one space variable and can be vectorized and/or parallelized quite easily. Furthermore, it is shown that a modification of the technique can be used to improve the properties of the underlying implicit schemes at little or no extra cost.