Application of the Complementary Derivatives Method in the ADI-FDTD Solution of Maxwell´s Equations

The alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is well suited for simulating structures with large aspect ratios or problems with large gradient fields where different grid sizes can be used to yield greater computational efficiency. However, using different grid sizes increases the truncation error at the interface between domains having different grid sizes. The truncation error is manifested as a spurious reflection from the grid boundary, thus decreasing the simulation accuracy. In this paper, we apply the complementary derivatives method (CDM) to reduce the spurious reflections arising from the use of different grid size domains when using the ADI-FDTD method. It is shown that, the CDM guarantees uniform second-order accuracy throughout the computational domain. When the CDM is implemented in the ADI-FDTD method, the implicit updating equations cannot be written in a tri-diagonal matrix and the computational efficiency of the ADI-FDTD method is not preserved. By employing the Sherman-Morrison formula, we retain the numerical efficiency of the conventional ADI-FDTD. A representative numerical example is presented to demonstrate the accuracy of CDM in the ADI-FDTD simulations.