**Memorandum 1762**

*The Gautschi time stepping scheme for edge finite element discretization of the Maxwell equations*

M.A. Botchev, D. Harutyunyan & J.J.W. van der Vegt

**Abstract:**

For the time integration of edge finite element
discretizations of the three-dimensional Maxwell
equations, we consider the Gautschi cosine scheme where
the action of the matrix function is approximated by
a Krylov subspace method.
First, for the space-discretized edge finite element Maxwell equations,
the dispersion error of this scheme is analyzed in detail
and compared to that of two conventional schemes.
Second, we show that the scheme can be implemented in such a way that
a higher accuracy can be achieved within less computational
time (as compared to other implicit schemes).
Although the new scheme is unconditionally stable, it is explicit in
structure: as an explicit scheme, it requires only the solution of linear
systems with the mass matrix.

**Keywords:**
Maxwell equations, Gautschi cosine scheme, dispersion analysis, edge elements, staggered leap frog scheme.

**Mathematics Subject Classification:**
35Q60, 78M10, 65M12, 37M15, 15A15

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