A Gas-Kinetic Scheme for Maxwell Equations

Abstract

The Gas-Kinetic Scheme (GKS), widely used in computational fluid dynamics for simulating hypersonic and other complicated flow phenomena, is extended in this work to electromagnetic problems by solving Maxwell's equations. In contrast to the classical GKS formulation, the proposed scheme employs a discrete rather than a continuous velocity space. By evaluating a time-accurate numerical flux at cell interfaces, the proposed scheme attains second-order accuracy within a single step. Its kinetic formulation provides an inherently multidimensional framework, while the finite-volume formulation ensures straightforward extension to unstructured meshes. Through the incorporation of a collision process, the scheme exhibits lower numerical dissipation than classical flux-vector splitting (FVS) methods. Furthermore, the kinetic decomposition enables direct implementation of non-reflecting boundary conditions. The proposed scheme is validated against several benchmark problems and compared with established methods, including the Finite-Difference Time-Domain (FDTD) method and FVS. A lattice Boltzmann method (LBM) implementation is also included for comparative analysis. Finally, the technique is applied to simulate electromagnetic wave propagation in a realistic aircraft configuration, demonstrating its ability to model complex geometries.

Figures (16)

• Figure 1: The lattice structure of D3Q4, D2Q4 and D1Q2.
• Figure 2: Plane wave test: Convergence order of GKS (left) and LBM (right).
• Figure 3: 2D TM modes: $E_z$ contour plots at $t=1.775s$.
• Figure 4: 2D TM modes: the $E_z$ profile at $y=0.3125$m (left) and the convergence order of GKS (right)
• Figure 5: Riemann problem: solution of the electromagnetic Riemann problem at $t=0.25$s comparing GKS, LBM, and FDTD methods. The GKS resolves discontinuities using a limiter. The second-order LBM and FDTD scheme produces expected oscillations near discontinuities.