An Explicit Noniterative Discontinuous Galerkin Time-Domain Method With Accurate Treatment for Kerr Nonlinearities

The interactions between electromagnetic (EM) waves and Kerr nonlinear media result in various appealing phenomena, including self-focusing, optical vortex soliton, etc. However, existing numerical methods for resolving Kerr nonlinearities demand multiple iterations or solutions to a huge number of cubic equations in each time advance, thus hardly achieving the optimal trade-off between accuracy and efficiency. In this article, an explicit non-iterative discontinuous Galerkin time-domain (DGTD) method is proposed for transient EM analysis involving Kerr media. The proposed method can be seamlessly integrated into the widely adopted Runge-Kutta-DGTD (RK-DGTD) method. The essence is the deliberate evaluation of the incremental electric field intensity, during which the only approximation required is to replace the current electric field intensity with that at the previous intermediate time instant of the RK scheme, so that implicit or iterative time stepping can be avoided. In addition, in regions with strong nonlinear effects or fine geometry, the computational burden is exacerbated by local fine meshes and the resultant tiny time-step size. Therefore, the local time-stepping (LTS) technique is further introduced to improve computational efficiency. Numerical results demonstrate that the proposed method can achieve high accuracy similar to those of Newton’s iterative method and direct method with the cubic formula, and consume roughly the same short execution time as that of the conventional explicit non-iterative method.