Sharp CFL stability and temporal-dispersion optimization of symmetric splitting schemes for time-domain Maxwell equations

Abstract

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 $s_{\ast }=12/(7\sqrt{d})$. 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 $6\sqrt{3}/(7\sqrt{d})$, 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.