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Title of Journal: Sci China Math

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Abbravation: Science China Mathematics

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Springer Berlin Heidelberg

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DOI

10.1016/0197-2456(93)90170-i

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1869-1862

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Optimal error estimates and modified energy conser

Authors: LiPing Gao Bo Zhang
Publish Date: 2013/03/20
Volume: 56, Issue: 8, Pages: 1705-1726
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Abstract

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finitedifference timedomain ADIFDTD method which is a popular scheme for solving the 3D Maxwell’s equations Precisely for the case with a perfectly electric conducting PEC boundary condition we establish the optimal secondorder error estimates in both space and time in the discrete H 1norm for the ADIFDTD scheme and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero then the discrete L 2norm of the discrete divergence of the ADIFDTD solution is approximately zero with the secondorder accuracy in both space and time The key ingredient is two new discrete modified energy norms which are secondorder in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper Furthermore we prove that in addition to two known discrete modified energy identities which are secondorder in time perturbations of two known energy conservation laws the ADIFDTD scheme also satisfies two new discrete modified energy identities which are secondorder in time perturbations of the two new energy conservation laws This means that the ADIFDTD scheme is unconditionally stable under the four discrete modified energy norms Experimental results which confirm the theoretical results are presented

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