时域麦克斯韦方程对称分裂格式的Sharp CFL稳定性与时域色散优化
Sharp CFL stability and temporal-dispersion optimization of symmetric splitting schemes for time-domain Maxwell equations
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中文总结 AI 辅助
本文针对时域麦克斯韦方程的显式回文电磁分裂格式，证明a=1/4是最大化谱CFL区间的唯一实系数，给出复共轭系数的相位精度阈值，通过实验区分稳定性与相位优化。
中文摘要 AI 辅助
我们分析了时域麦克斯韦方程的单参数族显式回文电磁分裂格式中的系数设计。在四阶交错空间离散后，傅里叶放大矩阵依赖于单个标量 $g_2=a(1-2a)/2$。我们证明 $a=1/4$ 是最大化谱CFL区间的唯一实系数，其阈值为 $s_*=12/(7\sqrt d)$。随后我们确定了高阶相位精度的实系数障碍：抵消主导时域相位缺陷要求 $g_2=1/12$，而所有实成员均满足 $g_2\le 1/16$。由此得到的复共轭系数对每个固定的半离散傅里叶模式给出四阶时域相位精度，阈值为 $6\sqrt3/(7\sqrt d)$，而完整场更新在时间上仍保持全局二阶精度。对于实麦克斯韦数据，物理输出是复轨迹的实投影；该投影与分支无关且保持二阶误差界。我们进一步给出完全等价的双精度实算术实现，其阐明了辅助虚分量的作用而不改变数值方法。半离散收敛结果与数值实验证实了稳定性优化与相位优化之间的区别。
英文摘要
We analyze coefficient design in a one-parameter family of explicit palindromic electric--magnetic splittings for the time-domain Maxwell equations. After fourth-order staggered spatial discretization, the Fourier amplification matrix depends on the single scalar $g_2=a(1-2a)/2$. We prove that $a=1/4$ is the unique real coefficient maximizing the spectral CFL interval, with threshold $s_*=12/(7\sqrt d)$. We then identify a real-coefficient obstruction to higher phase accuracy: cancellation of the leading temporal phase defect requires $g_2=1/12$, whereas every real member satisfies $g_2\le 1/16$. The resulting complex-conjugate coefficients give fourth-order temporal phase accuracy for each fixed semidiscrete Fourier mode and have threshold $6\sqrt3/(7\sqrt d)$, while the complete field update remains globally second order in time. For real Maxwell data, the physical output is the real projection of the complex trajectory; this projection is branch independent and preserves the second-order error bound. We further give an exactly equivalent doubled real-arithmetic realization, which clarifies the role of the auxiliary imaginary component without changing the numerical method. A semidiscrete convergence result and numerical experiments confirm the distinction between stability optimization and phase optimization.
