TD-EFIE with 3-Point Backward Difference and Stability Investigation Jaroslav LÁČÍK, Zbyněk RAIDA, Milan MOTL Dept. of Radio Engineering Brno University of Technology Purkyňova 118, CZ-61200 Brno CZECH REPUBLIC Abstract: In this paper, we present a new formulation using the time-domain electric field integral equation formulation (TD-EFIE) to obtain a transient scattering from conducting structures. The time derivative of the magnetic vector potential is approximated by 3-point backward difference. Further, the investigation of stability is presented. Key-Words: time-domain, method of moments, stability analysis. 1 Introduction Numerical techniques for the prediction of electro-magnetic fields scattered by complex objects, directly computing in time domain, have recently received considerable attention. Several formulations have been presented for the time-domain integral equation [1], [2]. In an EFIE, there is a time derivative of magnetic vector potential. By differentiating the EFIE with respect to time, this term is approximated by forward or backward finite differences, and an explicit or implicit solution has been presented [1]. Every solution, the explicit and implicit one, has advantages and disadvantages. By the explicit solution, an unknown current can be computed directly without solving an inverse matrix against the implicit one. However, the time step must satisfy Courant’s condition, the implicit one needn’t, and therefore the explicit one is time-consuming. The stability is the main problem of both approaches. The stability has been analyzed, and different interpretations have been formu- lated [3], [4]. E.g., a specific averaging tech- nique in time is proposed to filter out the late- time spurious fluctuations from the solutions in [3]. This method, however, does not generally provide criteria to make the numerical model unconditionally stable and it may sometimes imply a loss of accuracy in the solution. The explicit solution has stronger tendency to be- come unstable, and therefore we pay attention to the implicit one. In this work, we develop a new EFIE for- mulation with the 3-point backward difference, which provides accurate and stable solutions. We compare our proposed formulation to the others [1], [5]. Further, we investigate their stability with the help of algebraic analysis. 2 TD-EFIE Formulation Let S denote the surface of a closed or open perfect electric conducting (PEC) body illumi- nated by a transient electromagnetic wave. This incident wave induces a surface current J(r,t) on S. The scattered electric field E S (r,t) computed from the surface current is given by φ ∇ − ∂ ∂ − = t S A E , (1) where A and φ are the magnetic vector and the electric scalar potential defined as ∫ = S dS R t ' ) , ' ( 4 ) , ( τ π µ r J r A , (2)