Parallel-GPU DGTD method with a third-order local time stepping scheme

Abstract

The increasing use of numerical methods to solve large-scale and multiscale electromagnetic problems has driven the development of various strategies to enhance the efficiency of the Discontinuous Galerkin Time-Domain (DGTD) method without compromising accuracy. This work presents the combination of two such strategies aimed at improving the performance of the DGTD method and reducing execution time. The first strategy leverages Graphics Processing Units (GPUs) to accelerate computations by exploiting their low latency and high parallelism. The second employs a local time-stepping (LTS) technique, which allows different mesh elements to advance in time independently, thus avoiding the limitations imposed by a global time step (GTS). The study begins with a description of the spatial and temporal discretizations of the DGTD method. This is followed by an introduction to GPUs, highlighting their main characteristics and presenting an efficient data distribution scheme for executing DGTD computations. An LTS approach based on the third-order Runge-Kutta (RK3) method is then introduced, using a third-order polynomial to maintain accuracy. After developing both strategies, they are combined to form a more powerful and efficient numerical technique. To validate this approach, two-dimensional and three-dimensional electromagnetic problems are solved. Initial tests in homogeneous media, such as a metallic air-filled cavity, demonstrate the accuracy and performance of both shared and global GPU memory strategies, achieving speedups of up to 24× compared to CPU implementations. Further validation shows that the LTS-RK3 algorithm preserves numerical accuracy while reducing simulation time by up to 52% in an electromagnetic scattering problem when compared to the standard GTS approach. Finally, the combined strategy is applied to complex and multiscale problems, such as scattering by a multilayer sphere and radiation from a monopole antenna, achieving time reductions of nearly 78% and 55%, respectively. These results confirm that the proposed method significantly enhances computational performance while maintaining accuracy, outperforming the standard GTS implementation.