# Efficient Parallel FDTD Method Based on Non-Uniform Conformal Mesh
## 1 Introduction  
The FDTD method, pioneered by [PERSON] in 1966 [1], is well known as a seminal contribution in computational electromagnetics. Renowned for its succinct principle and ease of implementation, FDTD operates as a fully explicit algorithm devoid of matrix formulations, rendering it adopt at accommodating intricate structures and anisotropic materials. It has found extensive utility across diverse electromagnetic phenomena, including transmission, radiation, and scattering. One disadvantage of the traditional FDTD algorithm is that the use of uniform Cartesian mesh to discretize the computation domain results in a large number of grids. The non-uniform grid scheme [2] can simulate fine structures and reduce the number of grids. However, according to the Courant-Friedrichs-Lewy condition (CFL) [3], the maximum time step allowed is determined by the minimum mesh size of the computation domain, which means that the FDTD simulation must run a very large number of loops to reach a steady state. In addition, due to the step approximation adopted at the boundary of the model, not only will large errors be introduced, but sometimes false solutions will be obtained [4]. The mesh size needs to be reduced in order to reduce errors, which will inevitably increase the computational burden. To eliminate the problems caused by the step-wise approximation, [PERSON] and [PERSON] proposed a conformal finite-difference time-domain (CFDTD) method [5], in which the cut-cell mesh can better approximate arbitrary surfacesand accurately simulate complex geometry [6] with second-order computational accuracy [7]. The CFDTD method can obtain the same calculation accuracy as traditional methods by using fewer grids, which has the advantages of high precision and high efficiency, and has been widely used and researched [6; 7; 8; 9]. The combination of non-uniform and conformal mesh can greatly reduce the number of meshes and lower the amount of computation. Another limitation of traditional FDTD algorithms is the additional computational burden of applying a PML based on physical models or a truncation of simulation regions. A PML enables the waves transmitted to the truncation to be absorbed by the boundary without reflection, thus simulating the propagation of the electromagnetic field in infinite space. At present, the widely used uniaxial perfectly matched layer (UPML) [10] and convolutional perfectly matched layer (CPML) [11] are improvements of the original PML algorithm [12] proposed in the 1990s, which can more accurately simulate the propagation behavior of waves in complex media and improve the accuracy and reliability of simulation results. However, the application of a PML usually increases the number of grids and the amount of computation, which inevitably raises the computation burden. For FDTD methods based on non-uniform conformal grids and involving PML applications, the scale of problems that can be handled is severely limited by the available computing resources, and the rapid development of parallel computing provides solutions to the above problems. In particular, the FDTD method has a natural parallelism, which can solve large-scale problems on parallel platforms. For the moment, the parallel FDTD algorithm [13; 14; 15; 16; 17] has been widely implemented and tested on different computer platforms, among which the more commonly used field exchange method is to carry out one electric field exchange and one magnetic field exchange in the FDTD iteration process; the whole calculation process will be interrupted twice, and the processor will be idle and waiting when the electromagnetic field is transmitted. However, the field exchange method we adopted only needs one boundary magnetic field exchange while updating the internal electric field in the subregion, so that the transmission time and calculation time overlap, and the whole iteration process is interrupted only once, avoiding the waste of computing resources to a certain extent.