## 1. Introduction

[2] Nowadays, there is a widespread demand for high accuracy simulations needed to provide results which can parallel the accuracy of recent measurement equipments (e.g., 145 dB for R&S ZVA24 network analyzer from Rohde & Schwarz, and 136 dB for PNA series Microwave Network Analyzers from Agilent), can measure the shielding effectiveness up to 120 dB, or can measure the reflections from the absorbers that goes below 60 dB. When using the finite difference method [*Yee*, 1966; *Taflove and Hagness*, 2005; *Mitchell and Griffiths*, 1980], the simulation domain is discretized and then the differential equations are approximated with difference equations. The central finite difference is used very often to approximate the first-order differential equations and it results in second-order accuracy. Nonuniform grids are also used frequently, well suited for simulating structures with large aspect ratios or problems with large gradients, and yield greater computational efficiency. However, using different grid sizes increases the truncation error at the interface between domains having different grid sizes. The augment of truncation error at the interface is manifested as a spurious reflection from the grid boundary, thus decreasing the simulation accuracy.

[3] The grid size is chosen by considering two criteria: First, the grid size must resolve all structures in the simulation domain and, second, it must also resolve the highest frequency present in the simulation. Using a uniform grid suffers from flexibility when electric or magnetic fields (or both) have large gradients within a limited volume. For example, monolithic microwave integrated circuits (MMIC), in the vicinity of current sources, sharp edges and corners of conductive and dielectric objects, simulation of vias and bond wires in high-frequency electronic packages, and detailed, high dynamic range simulation in biomedical applications, need a very fine mesh to resolve the abrupt changes of fields in a very small volume. Considering the limitation of the computer memory and speed, a very fine mesh for the whole domain renders this method not very attractive for this large class of electromagnetic problems. A more computationally feasible solution would be to use refined mesh in certain subregions. The computational domain size is reduced by using different grid sizes in the simulation domain such that a coarse mesh is used for discretizing the domains with small field variations and a fine mesh is used for discretizing the domains with large field variations.

[4] Using different grid sizes improves the computational efficiency of the simulation, however, at the expense of increasing the truncation error at the interface between domains having different grid sizes. The truncation error at the interface is manifested as a spurious reflection from the grid boundary. Several papers have introduced different methods to reduce this spurious reflection or grid-truncation error [*Sheen*, 1991; *Lee*, 1990; *Li et al.*, 1993; *Xiao and Vahldieck*, 1993; *Jiang and Arai*, 1998; *Namiki and Ito*, 2001; *Okoniewski et al.*, 1997; *Heinrich et al.*, 1996]. In the work of *Sheen* [1991], the grid size in the second domain is reduced by one-third of the grid size in the first domain and spatial derivatives of the fields at the interface are expressed by central difference approximations to achieve second-order accuracy. However, in this method, the reduced grid size is limited to specific numbers which limits its applicability to specific geometries that conform to specialized grid. In the works of *Lee* [1990] and *Li et al.* [1993], the derivative of magnetic field at the electric field position is approximated by fitting a second-degree polynomial to the magnetic fields at three points. The coefficient in the error term of this approximation, however, is large, which limits the grid size reduction factor. In the work of *Xiao and Vahldieck* [1993], two methods were introduced to maintain the second-order accuracy. One method uses a specific mesh ratio between two regions to obtain the central finite differences, while the second uses a universal grading scheme with continuously variable lattice size, however, a demonstration of the performance of these method were not reported. In the work of *Jiang and Arai* [1998], the computational accuracy was improved by interpolating the magnetic field components between the fine mesh and coarse mesh, which cannot guarantee the second-order accuracy. In the work of *Namiki and Ito* [2001], a high-order implicit scheme was enforced at the boundary to reduce the truncation error. In the work of *Okoniewski et al.* [1997], a numerically derived three-dimensional subgridding scheme was introduced but without theoretical limits on its potential. In the work of *Heinrich et al.* [1996], the characteristic impedances of two waveguides with infinitely thin and square center conductors are calculated using conformal mapping and mode matching, respectively. Then, the values of minimum grid size and amplification factor were optimized to obtain good approximations of the characteristic impedances using the FDTD method. But there is no explanation whether these values can also be used for other experiments or whether each experiment needs different optimized values, thus limiting the generality and practicality of this method. More recently, two new methods were presented by *Bérenger* [2006] and *Pierantoni and Rozzi* [2007] that were based on Huygens' equivalent currents and Hilbert Space formulations, respectively. While these methods do not suffer from the constraints of earlier methods, the error bounds arising from the nonuniform grid domains have yet to be investigated.

[5] In this paper, first we review the theory of the complementary derivatives method (CDM) [*Kermani and Ramahi*, 2006a, 2006b, 2005], we then present a full analytical investigation of the method for the one-dimensional case. We derive the fundamental modes of propagation in the numerical solution of the differential equation and calculate the reflection coefficient from the interface of two domains having different grid sizes when CDM is applied. The performance of the CDM is demonstrated by showing a comparison between the reflection coefficients for the numerical solution of the wave equation, calculated with and without the CDM.