Microlocal diagonalization of strictly hyperbolic pseudodifferential systems and application to the design of radiation conditions in electromagnetism

In [Comm. Pure Appl. Math., 28 ( 1975), pp. 457-478], M. E. Taylor describe s a constructive diagonalization method to investigate the reflection of si ngularities of the solution to first-order hyperbolic systems. According to further works initiated by Engquist and Majda, this approach proved to be well adapted to the construction of artificial boundary conditions. However , in the case of systems governed by pseudodifferential operators with vari able coefficients, Taylor's method involves very elaborate calculations of the symbols of the operators. Hence, a direct approach may be di cult when dealing with high-order conditions. This motivates the rst part of this pap er, where a recursive explicit formulation of Taylor's method is stated for microlocally strictly hyperbolic systems. Consequently, it provides an alg orithm leading to symbolic calculations which can be handled by a computer algebra system. In the second part, an application of the method is investi gated for the construction of local radiation boundary conditions on arbitr arily shaped surfaces for the transverse electric Maxwell system. It is pro ved that they are of complete order, provided the introduction of a new sym bols class well adapted to the Maxwell system. Next, a complete second-orde r condition is designed, and when used as an on-surface radiation condition [ G. A. Kriegsmann, A. Taflove, and K. R. Umashankar, IEEE Trans. Antennas and Propagation, 35 (1987), pp. 153-161], it can give rise to an ultrafast numerical approximate solution of the scattered field.