Higher Order Leapfrog Schemes

According to the leapfrog concept, the eAAt quantity is approximated via Taylor expansion as 

If higher order accuracy is required, the suitable number of additional terms in (5.58) should be considered. Hence, the general time-marching equations for 2k, M) FDTD schemes 

In other words, the only difference between (5.60) and the second-order leapfrog scheme is of the supplementary terms - depending on the value of k - which increase the computational overhead with the consecutive use of spatial operators at additional nodes. It is stressed that the presence of lossy materials receives an analogous treatment with the exception of more complicated expressions beyond the fourth order. 

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H. Spachmann, R. Schuhmann, and T. Weiland, Convergence, stability and dispersion analysis of higher order leapfrog schemes for Maxwell s equations, in Proc. 17th Ann. Rev. Prog. Appl. Comput. Electromagn., Monterey, CA, Mar. 2001, pp. 655-662. 

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