Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of \(10^{-6}\).

几何光学中关于相空间的刘维尔方程描述了通过光学系统的能量分布的演变，该系统在光学界面上是不连续的。不连续伽辽金谱元法比较保守，可以在局部实现更高阶的收敛，使其成为该方程的合适方法。在处理相空间中的光学界面时，会出现非局部边界条件。除了本身是一个困难之外，这些非局部边界条件还必须满足能量守恒约束。为此，我们引入了光学界面的能量保守处理。数值实验证明该方法遵循能量守恒。此外，该方法与行业标准的光线追踪进行了比较。\(10^{-6}\)。