Cartesian Yee symbol, numerical dispersion, and grid CFL bound

On a periodic Fourier mode, a forward edge difference in direction $j$ has multiplier $e^{i{\theta }_j}-1=2i{e}^{i{\theta }_j\mathrm{/}2}\sin ({\theta }_j\mathrm{/}2)$. The half-cell phase in the edge and face cochain locations is removed by a unitary diagonal change of Fourier basis, so the scaled directional symbol is $i{d}_j$ with $d_j=2\sin ({\theta }_j\mathrm{/}2)\mathrm{/}{h}_j$. The Cartesian curl symbol is therefore $i\mathit{d}\times$.

The cross-product matrix $\mathit{d}\times$ is real skew-symmetric. The identity $-(\mathit{d}\times {)}^2=\mathrm{\mid }\mathit{d}{\mathrm{\mid }}^{2}I-\mathit{d}{\mathit{d}}^T$ follows by expanding the vector triple product. Thus the curl-curl symbol is zero on the longitudinal vector $\mathit{d}$ and equals $\mathrm{\mid }\mathit{d}{\mathrm{\mid }}^2$ on its two-dimensional orthogonal complement. The Hodge weights in [F1] scale this eigenvalue by $c^2$, giving the stated $\lambda (\theta )$.

For a transverse mode $e^n=q^{n}v$ with this eigenvalue, subtracting the electric row in [F3] at consecutive time indices and using its magnetic row gives $q^{n+1}-2q^n+q^{n-1}+\mathrm{\Delta }{t}^2\lambda (\theta )q^n=0$. Substitution of $q^n=e^{-i\omega n\mathrm{\Delta }t}$ yields $4{\sin }^2(\omega \mathrm{\Delta }t\mathrm{/}2)=\mathrm{\Delta }{t}^2\lambda (\theta )$, which is the displayed dispersion relation.

Since $0\le {\sin }^2({\theta }_j\mathrm{/}2)\le 1$, $\lambda (\theta )\le 4c^2(h_x^{-2}+h_y^{-2}+h_z^{-2})$. When every $N_j$ is even, the allowed tuple $\theta =(\pi ,\pi ,\pi )$ attains equality; otherwise the exact maximum is the maximum over the finite allowed tuples. Applying [F2] gives the rectangular strict bound and its cubic specialization, while preserving the finite-grid endpoint distinction.