In this paper we investigate an alternating direction implicit (ADI) time integration scheme for the linear Maxwell equations with currents, charges and conductivity. We show its stability and efficiency. The main results establish that the scheme converges in a space similar to H −1 with order two to the solution of the Maxwell system. Moreover, the divergence conditions in the system are preserved in H −1 with order one.

Abstract. We investigate the Lie and the Strang splitting for the cubic nonlinear Schrödinger equation on the full space and on the torus in up to three spatial dimensions. We prove that the Strang splitting converges in L 2 with order 1 + θ for initial values in H 2+2θ with θ ∈ (0, 1) and that the Lie splitting converges with order one for initial values in H 2 .

We investigate an alternating direction implicit (ADI) scheme for the time-integration of the Maxwell equations with currents, charges and conductivity. This method is unconditionally stable, numerically efficient, and preserves the norm of the solution exactly in absence of the external current and the conductivity. We prove that the semidiscretization in time converges in a space similar to H −1 with order two to the solution of the Maxwell system.2010 Mathematics Subject Classification. 65M12; 35Q61, 47D06, 65J10.

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