Finite-difference Green's functions on a 3-D cubic lattice — Integer versus fixed-precision arithmetic recurrence schemes
Time-domain 3-D lattice Green's function (LGF) sequences can be evaluated using a single-lattice point recurrence scheme, and play an important role in finite-difference Green's function diakoptics. Asymptotically, at large distances, the LGFs in three dimensions can be described in terms of six wave constituents, each oscillating with its own instantaneous complex or real frequency. All instantaneous frequencies eventually become real, say at discrete time nAC. We present evidence that indicates that if fixed-precision arithmetic results in a prohibitive loss of accuracy, then that happens before nAC. If a sufficient number of significant digits is used, and loss of accuracy is curtailed before n = nAC, then the recurrence scheme will yield reliable results for a long time after (if not indefinitely). Moreover, the resulting fixed-precision recurrence scheme is still considerably more efficient than the exact scheme based on integer arithmetic.