Acceleration of the Discrete Green’s Function Formulation of the FDTD Method Based on Recurrence Schemes

In this paper, we investigate an acceleration of the discrete Green's function (DGF) formulation of the FDTD method (DGF-FDTD) with the use of recurrence schemes. The DGF-FDTD method allows one to compute FDTD solutions as a convolution of the excitation with the DGF kernel. Hence, it does not require to execute a leapfrog time-stepping scheme in a whole computational domain for this purpose. Until recently, the DGF generation has been the limiting step of DGF-FDTD due to large computational resources, in terms of processor time and memory, required for these computations. Hence, we have derived the no-neighbours recurrence scheme for one-dimensional FDTD-compatible DGF using solely properties of the Gauss hypergeometric function (GHF). Using known properties of GHF, the recurrence scheme is obtained for arbitrary stable time-step size. In this paper, we show that using the recurrence scheme, computations of 1-D FDTD solutions with the use of the DGF-FDTD method can be around an order of magnitude faster than those based on the direct FDTD method. Although 2- and 3-D recurrence schemes for DGF (valid not only for the magic time-step size) still need to be derived, the 1-D case remains the starting point for any research in this area.