Time domain simulation of electromagnetic fields is of great practical significance in engineering and physics, since it allows a broad spectrum of frequencies to be analyzed in a single simulation. The classical central difference scheme introduced by K. Yee (1966) has been extensively tested in a wide range of applications, and its capabilities have been established. Meanwhile, as is always the case in a second order method (in both space and time), the Yee scheme begins to accumulate phase errors as time grows large, especially for electrically large domains or for late-time analysis. Because of this, a number of attempts have been made to extend this scheme to fourth order accuracy since J. Fang's pioneering work (1989). Fourth order methods allow a coarser time step and spatial mesh to be used while maintaining the same accuracy (effectively increasing simulation speed), or can increase the accuracy for a given mesh spacing and time step.

Unfortunately, there are many well-known difficulties in implementing a fourth order scheme for the Maxwell equations with a physical boundary condition. These include instability near the boundary or the requirement of an implicit updating scheme. In a joint work with my collaborators from the Department of Electrical Engineering of University of Tennessee and Oak Ridge National Lab, we proposed and analyzed a fully explicit fourth order scheme over the Yee mesh grid. A careful treatment of the numerical values near the boundary leads to a ``symmetric image'' formula at the ``ghost'' grid points, which does not suffer from an instability. To advance the dynamic equations, the Jameson method, an alternate four-stage integrator is utilized. The main advantage of this alternate method over the classical formula is the memory requirement reduction in the numerical implementation, since only the numerical profile at the previous time stage is needed in the time integration. Such a time integration was proposed by A. Jameson in the numerical simulations of gas dynamics in the 1980s. However, this method was not much appreciated in the community of scientific computing, since it is only second order accurate for nonlinear equations.

We applied the proposed scheme to compute the fields in lossless rectangular cavity. Different modes are included in the initial data with a decaying magnitude and an FFT (fast Fourier transformation) tool is utilized to analyze the frequency of the wave over time. The numerical results indicate that the proposed scheme performs much better than the classical second order methods in the accuracy of resonant frequency computations.

In addition, we have extended the scheme to a more general computational domain and physical setup. Right now, we are working on the simulation of a ridge-shape cavity. This project has potential applications in telecommunications and waveguide filters, and the numerical results are expected to be submitted to an engineering journal. Due to its geometric complexity, more complicated structures of the solution have been observed.

I would like to remark that such an extension of my research direction to computational electro-magnetics is largely based on my previous work on the computation of fluid dynamics. With some careful modifications, a similar methodology can be applied to treat the Maxwell equations, which yields very robust numerical results compared to conventional numerical methods and commonly-used commercial softwares.

As a particular example, a detailed numerical simulation of a twisted waveguide was performed. This project has potential applications in the design of a novel accelerator cavity. A twisted waveguide would be cheaper to manufacture at a massive scale than a conventional accelerator cavity. Such an engineering design was initiated by Y. Kang (in 2000) and a more detailed study was undertaken by J. Wilson. Meanwhile, a detailed numerical simulation has to be performed before the manufacturing process can proceed, to ensure its stability and related properties. However, a careful numerical experiment shows that the conventional commercial software fails in the numerical simulation of a waveguide with a high twist rate (which is needed for the practical application). Moreover, there is no known numerical algorithm which can work well for this problem.