Use raw strings to avoid double backslashes
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@ -1,4 +1,4 @@
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"""
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r"""
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Tools for finite difference frequency-domain (FDFD) simulations and calculations.
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These mostly involve picking a single frequency, then setting up and solving a
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@ -19,71 +19,71 @@ Submodules:
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From the "Frequency domain" section of `meanas.fdmath`, we have
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$$
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\\begin{aligned}
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\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
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\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{H}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
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\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
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\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{M}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega l \\Delta_t} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
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-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\
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\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
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\\end{aligned}
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\begin{aligned}
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\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
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-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
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\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
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\end{aligned}
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$$
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resulting in
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$$
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\\begin{aligned}
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\\tilde{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
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\\hat{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
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\\end{aligned}
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\begin{aligned}
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\tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
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\hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
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\end{aligned}
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$$
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Maxwell's equations are then
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$$
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\\begin{aligned}
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\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
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\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2} \\hat{B}_{\\vec{r} + \\frac{1}{2}}
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- \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
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-\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2} \\tilde{D}_{\\vec{r}}
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+ \\tilde{J}_{\\vec{r}} \\\\
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\\tilde{\\nabla} \\cdot \\hat{B}_{\\vec{r} + \\frac{1}{2}} &= 0 \\\\
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\\hat{\\nabla} \\cdot \\tilde{D}_{\\vec{r}} &= \\rho_{\\vec{r}}
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\\end{aligned}
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\begin{aligned}
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}}
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- \hat{M}_{\vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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-\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}}
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+ \tilde{J}_{\vec{r}} \\
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\tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\
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\hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}}
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\end{aligned}
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$$
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With $\\Delta_t \\to 0$, this simplifies to
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With $\Delta_t \to 0$, this simplifies to
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$$
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\\begin{aligned}
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\\tilde{E}_{l, \\vec{r}} &\\to \\tilde{E}_{\\vec{r}} \\\\
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\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{H}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\tilde{J}_{l, \\vec{r}} &\\to \\tilde{J}_{\\vec{r}} \\\\
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\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{M}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\Omega &\\to \\omega \\\\
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\\tilde{\\partial}_t &\\to -\\imath \\omega \\\\
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\\hat{\\partial}_t &\\to -\\imath \\omega \\\\
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\\end{aligned}
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\begin{aligned}
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\tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\
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\tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\
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\Omega &\to \omega \\
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\tilde{\partial}_t &\to -\imath \omega \\
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\hat{\partial}_t &\to -\imath \omega \\
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\end{aligned}
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$$
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and then
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$$
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\\begin{aligned}
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\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
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\\imath \\omega \\hat{B}_{\\vec{r} + \\frac{1}{2}}
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- \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
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-\\imath \\omega \\tilde{D}_{\\vec{r}}
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+ \\tilde{J}_{\\vec{r}} \\\\
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\\end{aligned}
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\begin{aligned}
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
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- \hat{M}_{\vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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-\imath \omega \tilde{D}_{\vec{r}}
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+ \tilde{J}_{\vec{r}} \\
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\end{aligned}
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$$
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$$
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
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-\\omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\omega \\tilde{J}_{\\vec{r}} \\\\
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
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-\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\
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$$
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# TODO FDFD?
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@ -189,10 +189,10 @@ def e_tfsf_source(
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def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], cfdfield_t]:
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"""
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r"""
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Generates a function that takes the single-frequency `E` and `H` fields
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and calculates the cross product `E` x `H` = $E \\times H$ as required
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for the Poynting vector, $S = E \\times H$
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and calculates the cross product `E` x `H` = $E \times H$ as required
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for the Poynting vector, $S = E \times H$
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Note:
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This function also shifts the input `E` field by one cell as required
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@ -45,14 +45,14 @@ def e_full(
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pec: vfdfield_t | None = None,
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pmc: vfdfield_t | None = None,
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) -> sparse.spmatrix:
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"""
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r"""
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Wave operator
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$$ \\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon $$
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$$ \nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon $$
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del x (1/mu * del x) - omega**2 * epsilon
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for use with the E-field, with wave equation
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$$ (\\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon) E = -\\imath \\omega J $$
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$$ (\nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon) E = -\imath \omega J $$
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(del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
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@ -131,14 +131,14 @@ def h_full(
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pec: vfdfield_t | None = None,
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pmc: vfdfield_t | None = None,
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) -> sparse.spmatrix:
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"""
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r"""
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Wave operator
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$$ \\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu $$
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$$ \nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu $$
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del x (1/epsilon * del x) - omega**2 * mu
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for use with the H-field, with wave equation
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$$ (\\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu) E = \\imath \\omega M $$
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$$ (\nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu) E = \imath \omega M $$
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(del x (1/epsilon * del x) - omega**2 * mu) E = i * omega * M
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@ -188,28 +188,28 @@ def eh_full(
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pec: vfdfield_t | None = None,
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pmc: vfdfield_t | None = None,
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) -> sparse.spmatrix:
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"""
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r"""
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Wave operator for `[E, H]` field representation. This operator implements Maxwell's
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equations without cancelling out either E or H. The operator is
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$$ \\begin{bmatrix}
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-\\imath \\omega \\epsilon & \\nabla \\times \\\\
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\\nabla \\times & \\imath \\omega \\mu
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\\end{bmatrix} $$
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$$ \begin{bmatrix}
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-\imath \omega \epsilon & \nabla \times \\
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\nabla \times & \imath \omega \mu
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\end{bmatrix} $$
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[[-i * omega * epsilon, del x ],
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[del x, i * omega * mu]]
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for use with a field vector of the form `cat(vec(E), vec(H))`:
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$$ \\begin{bmatrix}
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-\\imath \\omega \\epsilon & \\nabla \\times \\\\
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\\nabla \\times & \\imath \\omega \\mu
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\\end{bmatrix}
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\\begin{bmatrix} E \\\\
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H
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\\end{bmatrix}
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= \\begin{bmatrix} J \\\\
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-M
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\\end{bmatrix} $$
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$$ \begin{bmatrix}
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-\imath \omega \epsilon & \nabla \times \\
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\nabla \times & \imath \omega \mu
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\end{bmatrix}
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\begin{bmatrix} E \\
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H
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\end{bmatrix}
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= \begin{bmatrix} J \\
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-M
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\end{bmatrix} $$
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Args:
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omega: Angular frequency of the simulation
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@ -211,7 +211,7 @@ def operator_e(
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$$
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\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\
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0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
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0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
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\begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\
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\mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1}
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\begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
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@ -1,4 +1,4 @@
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"""
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r"""
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Basic discrete calculus for finite difference (fd) simulations.
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@ -43,11 +43,11 @@ Scalar derivatives and cell shifts
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----------------------------------
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Define the discrete forward derivative as
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$$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
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$$ [\tilde{\partial}_x f]_{m + \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m + 1} - f_m) $$
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where $f$ is a function defined at discrete locations on the x-axis (labeled using $m$).
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The value at $m$ occupies a length $\\Delta_{x, m}$ along the x-axis. Note that $m$
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The value at $m$ occupies a length $\Delta_{x, m}$ along the x-axis. Note that $m$
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is an index along the x-axis, _not_ necessarily an x-coordinate, since each length
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$\\Delta_{x, m}, \\Delta_{x, m+1}, ...$ is independently chosen.
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$\Delta_{x, m}, \Delta_{x, m+1}, ...$ is independently chosen.
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If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]`
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along the x-axis, the forward derivative is
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@ -56,13 +56,13 @@ along the x-axis, the forward derivative is
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Likewise, discrete reverse derivative is
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$$ [\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
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$$ [\hat{\partial}_x f ]_{m - \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
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or
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deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
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The derivatives' values are shifted by a half-cell relative to the original function, and
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will have different cell widths if all the `dx[i]` ( $\\Delta_{x, m}$ ) are not
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will have different cell widths if all the `dx[i]` ( $\Delta_{x, m}$ ) are not
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identical:
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[figure: derivatives and cell sizes]
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@ -88,19 +88,19 @@ identical:
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In the above figure,
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`f0 =` $f_0$, `f1 =` $f_1$
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`Df0 =` $[\\tilde{\\partial}f]_{0 + \\frac{1}{2}}$
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`Df1 =` $[\\tilde{\\partial}f]_{1 + \\frac{1}{2}}$
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`df0 =` $[\\hat{\\partial}f]_{0 - \\frac{1}{2}}$
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`Df0 =` $[\tilde{\partial}f]_{0 + \frac{1}{2}}$
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`Df1 =` $[\tilde{\partial}f]_{1 + \frac{1}{2}}$
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`df0 =` $[\hat{\partial}f]_{0 - \frac{1}{2}}$
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etc.
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The fractional subscript $m + \\frac{1}{2}$ is used to indicate values defined
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The fractional subscript $m + \frac{1}{2}$ is used to indicate values defined
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at shifted locations relative to the original $m$, with corresponding lengths
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$$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
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$$ \Delta_{x, m + \frac{1}{2}} = \frac{1}{2} * (\Delta_{x, m} + \Delta_{x, m + 1}) $$
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Just as $m$ is not itself an x-coordinate, neither is $m + \\frac{1}{2}$;
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Just as $m$ is not itself an x-coordinate, neither is $m + \frac{1}{2}$;
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carefully note the positions of the various cells in the above figure vs their labels.
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If the positions labeled with $m$ are considered the "base" or "original" grid,
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the positions labeled with $m + \\frac{1}{2}$ are said to lie on a "dual" or
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the positions labeled with $m + \frac{1}{2}$ are said to lie on a "dual" or
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"derived" grid.
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For the remainder of the `Discrete calculus` section, all figures will show
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@ -113,12 +113,12 @@ Gradients and fore-vectors
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--------------------------
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Expanding to three dimensions, we can define two gradients
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$$ [\\tilde{\\nabla} f]_{m,n,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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\\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
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\\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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$$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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\\vec{y} [\\hat{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
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\\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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$$ [\tilde{\nabla} f]_{m,n,p} = \vec{x} [\tilde{\partial}_x f]_{m + \frac{1}{2},n,p} +
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\vec{y} [\tilde{\partial}_y f]_{m,n + \frac{1}{2},p} +
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\vec{z} [\tilde{\partial}_z f]_{m,n,p + \frac{1}{2}} $$
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$$ [\hat{\nabla} f]_{m,n,p} = \vec{x} [\hat{\partial}_x f]_{m + \frac{1}{2},n,p} +
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\vec{y} [\hat{\partial}_y f]_{m,n + \frac{1}{2},p} +
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\vec{z} [\hat{\partial}_z f]_{m,n,p + \frac{1}{2}} $$
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or
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@ -144,12 +144,12 @@ y in y, and z in z.
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We call the resulting object a "fore-vector" or "back-vector", depending
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on the direction of the shift. We write it as
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$$ \\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p} +
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\\vec{y} g^y_{m,n + \\frac{1}{2},p} +
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\\vec{z} g^z_{m,n,p + \\frac{1}{2}} $$
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$$ \\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} +
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\\vec{y} g^y_{m,n - \\frac{1}{2},p} +
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\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
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$$ \tilde{g}_{m,n,p} = \vec{x} g^x_{m + \frac{1}{2},n,p} +
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\vec{y} g^y_{m,n + \frac{1}{2},p} +
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\vec{z} g^z_{m,n,p + \frac{1}{2}} $$
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$$ \hat{g}_{m,n,p} = \vec{x} g^x_{m - \frac{1}{2},n,p} +
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\vec{y} g^y_{m,n - \frac{1}{2},p} +
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\vec{z} g^z_{m,n,p - \frac{1}{2}} $$
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[figure: gradient / fore-vector]
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@ -172,15 +172,15 @@ Divergences
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There are also two divergences,
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$$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
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= [\\tilde{\\partial}_x g^x]_{m,n,p} +
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[\\tilde{\\partial}_y g^y]_{m,n,p} +
|
||||
[\\tilde{\\partial}_z g^z]_{m,n,p} $$
|
||||
$$ d_{n,m,p} = [\tilde{\nabla} \cdot \hat{g}]_{n,m,p}
|
||||
= [\tilde{\partial}_x g^x]_{m,n,p} +
|
||||
[\tilde{\partial}_y g^y]_{m,n,p} +
|
||||
[\tilde{\partial}_z g^z]_{m,n,p} $$
|
||||
|
||||
$$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p}
|
||||
= [\\hat{\\partial}_x g^x]_{m,n,p} +
|
||||
[\\hat{\\partial}_y g^y]_{m,n,p} +
|
||||
[\\hat{\\partial}_z g^z]_{m,n,p} $$
|
||||
$$ d_{n,m,p} = [\hat{\nabla} \cdot \tilde{g}]_{n,m,p}
|
||||
= [\hat{\partial}_x g^x]_{m,n,p} +
|
||||
[\hat{\partial}_y g^y]_{m,n,p} +
|
||||
[\hat{\partial}_z g^z]_{m,n,p} $$
|
||||
|
||||
or
|
||||
|
||||
@ -203,7 +203,7 @@ where `g = [gx, gy, gz]` is a fore- or back-vector field.
|
||||
|
||||
Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
|
||||
is defined at the back-vector's (fore-vector's) location $(m,n,p)$ and not at the locations of its components
|
||||
$(m \\pm \\frac{1}{2},n,p)$ etc.
|
||||
$(m \pm \frac{1}{2},n,p)$ etc.
|
||||
|
||||
[figure: divergence]
|
||||
^^
|
||||
@ -227,23 +227,23 @@ Curls
|
||||
|
||||
The two curls are then
|
||||
|
||||
$$ \\begin{aligned}
|
||||
\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
|
||||
[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
|
||||
\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
|
||||
&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
|
||||
&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
|
||||
\\end{aligned} $$
|
||||
$$ \begin{aligned}
|
||||
\hat{h}_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &= \\
|
||||
[\tilde{\nabla} \times \tilde{g}]_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &=
|
||||
\vec{x} (\tilde{\partial}_y g^z_{m,n,p + \frac{1}{2}} - \tilde{\partial}_z g^y_{m,n + \frac{1}{2},p}) \\
|
||||
&+ \vec{y} (\tilde{\partial}_z g^x_{m + \frac{1}{2},n,p} - \tilde{\partial}_x g^z_{m,n,p + \frac{1}{2}}) \\
|
||||
&+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^z_{m + \frac{1}{2},n,p})
|
||||
\end{aligned} $$
|
||||
|
||||
and
|
||||
|
||||
$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
|
||||
[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
|
||||
$$ \tilde{h}_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} =
|
||||
[\hat{\nabla} \times \hat{g}]_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} $$
|
||||
|
||||
where $\\hat{g}$ and $\\tilde{g}$ are located at $(m,n,p)$
|
||||
with components at $(m \\pm \\frac{1}{2},n,p)$ etc.,
|
||||
while $\\hat{h}$ and $\\tilde{h}$ are located at $(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$
|
||||
with components at $(m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ etc.
|
||||
where $\hat{g}$ and $\tilde{g}$ are located at $(m,n,p)$
|
||||
with components at $(m \pm \frac{1}{2},n,p)$ etc.,
|
||||
while $\hat{h}$ and $\tilde{h}$ are located at $(m \pm \frac{1}{2}, n \pm \frac{1}{2}, p \pm \frac{1}{2})$
|
||||
with components at $(m, n \pm \frac{1}{2}, p \pm \frac{1}{2})$ etc.
|
||||
|
||||
|
||||
[code: curls]
|
||||
@ -287,27 +287,27 @@ Maxwell's Equations
|
||||
|
||||
If we discretize both space (m,n,p) and time (l), Maxwell's equations become
|
||||
|
||||
$$ \\begin{aligned}
|
||||
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
|
||||
- \\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
|
||||
+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
||||
\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
|
||||
\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
|
||||
\\end{aligned} $$
|
||||
$$ \begin{aligned}
|
||||
\tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}
|
||||
- \hat{M}_{l, \vec{r} + \frac{1}{2}} \\
|
||||
\hat{\nabla} \times \hat{H}_{l-\frac{1}{2},\vec{r} + \frac{1}{2}} &= \hat{\partial}_t \tilde{D}_{l, \vec{r}}
|
||||
+ \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
|
||||
\tilde{\nabla} \cdot \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} &= 0 \\
|
||||
\hat{\nabla} \cdot \tilde{D}_{l,\vec{r}} &= \rho_{l,\vec{r}}
|
||||
\end{aligned} $$
|
||||
|
||||
with
|
||||
|
||||
$$ \\begin{aligned}
|
||||
\\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}}
|
||||
\\end{aligned} $$
|
||||
$$ \begin{aligned}
|
||||
\hat{B}_{\vec{r}} &= \mu_{\vec{r} + \frac{1}{2}} \cdot \hat{H}_{\vec{r} + \frac{1}{2}} \\
|
||||
\tilde{D}_{\vec{r}} &= \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}}
|
||||
\end{aligned} $$
|
||||
|
||||
where the spatial subscripts are abbreviated as $\\vec{r} = (m, n, p)$ and
|
||||
$\\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})$,
|
||||
$\\tilde{E}$ and $\\hat{H}$ are the electric and magnetic fields,
|
||||
$\\tilde{J}$ and $\\hat{M}$ are the electric and magnetic current distributions,
|
||||
and $\\epsilon$ and $\\mu$ are the dielectric permittivity and magnetic permeability.
|
||||
where the spatial subscripts are abbreviated as $\vec{r} = (m, n, p)$ and
|
||||
$\vec{r} + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2})$,
|
||||
$\tilde{E}$ and $\hat{H}$ are the electric and magnetic fields,
|
||||
$\tilde{J}$ and $\hat{M}$ are the electric and magnetic current distributions,
|
||||
and $\epsilon$ and $\mu$ are the dielectric permittivity and magnetic permeability.
|
||||
|
||||
The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
|
||||
The time derivatives can be expanded to form the update equations:
|
||||
@ -369,34 +369,34 @@ Each component forms its own grid, offset from the others:
|
||||
|
||||
The divergence equations can be derived by taking the divergence of the curl equations
|
||||
and combining them with charge continuity,
|
||||
$$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$
|
||||
$$ \hat{\nabla} \cdot \tilde{J} + \hat{\partial}_t \rho = 0 $$
|
||||
implying that the discrete Maxwell's equations do not produce spurious charges.
|
||||
|
||||
|
||||
Wave equation
|
||||
-------------
|
||||
|
||||
Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and
|
||||
replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation,
|
||||
and setting $\\hat{M}$ to zero, we can form the discrete wave equation:
|
||||
Taking the backward curl of the $\tilde{\nabla} \times \tilde{E}$ equation and
|
||||
replacing the resulting $\hat{\nabla} \times \hat{H}$ term using its respective equation,
|
||||
and setting $\hat{M}$ to zero, we can form the discrete wave equation:
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
|
||||
-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
|
||||
- \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
|
||||
-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
||||
\\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}) \\\\
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
||||
-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
||||
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
|
||||
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
|
||||
&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
|
||||
-\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}
|
||||
- \hat{M}_{l-1, \vec{r} + \frac{1}{2}} \\
|
||||
\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
|
||||
-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\
|
||||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
|
||||
\hat{\nabla} \times (-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}) \\
|
||||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
|
||||
-\tilde{\partial}_t \hat{\nabla} \times \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\
|
||||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
|
||||
-\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} + \hat{\partial}_t \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
|
||||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}})
|
||||
+ \tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \cdot \tilde{E}_{l, \vec{r}}
|
||||
&= \tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}}
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
|
||||
@ -406,27 +406,27 @@ Frequency domain
|
||||
We can substitute in a time-harmonic fields
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
|
||||
\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
|
||||
\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t}
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
resulting in
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
|
||||
\\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
|
||||
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\tilde{\partial}_t &\Rightarrow (e^{ \imath \omega \Delta_t} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{-\imath \omega \Delta_t / 2} = -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
|
||||
\hat{\partial}_t &\Rightarrow (1 - e^{-\imath \omega \Delta_t}) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{ \imath \omega \Delta_t / 2} = -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
|
||||
\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
This gives the frequency-domain wave equation,
|
||||
|
||||
$$
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
|
||||
-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\
|
||||
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
|
||||
-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
|
||||
$$
|
||||
|
||||
|
||||
@ -436,48 +436,48 @@ Plane waves and Dispersion relation
|
||||
With uniform material distribution and no sources
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
|
||||
\\epsilon_{\\vec{r}} &= \\epsilon \\\\
|
||||
\\tilde{J}_{\\vec{r}} &= 0 \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\mu_{\vec{r} + \frac{1}{2}} &= \mu \\
|
||||
\epsilon_{\vec{r}} &= \epsilon \\
|
||||
\tilde{J}_{\vec{r}} &= 0 \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
the frequency domain wave equation simplifies to
|
||||
|
||||
$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
|
||||
$$ \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}} - \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 $$
|
||||
|
||||
Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify
|
||||
Since $\hat{\nabla} \cdot \tilde{E}_{\vec{r}} = 0$, we can simplify
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
|
||||
&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
|
||||
&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
|
||||
&= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}}
|
||||
&= \tilde{\nabla}(\hat{\nabla} \cdot \tilde{E}_{\vec{r}}) - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\
|
||||
&= - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\
|
||||
&= - \tilde{\nabla}^2 \tilde{E}_{\vec{r}}
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
and we get
|
||||
|
||||
$$ \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} + \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
|
||||
$$ \tilde{\nabla}^2 \tilde{E}_{\vec{r}} + \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 $$
|
||||
|
||||
We can convert this to three scalar-wave equations of the form
|
||||
|
||||
$$ (\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0 $$
|
||||
$$ (\tilde{\nabla}^2 + K^2) \phi_{\vec{r}} = 0 $$
|
||||
|
||||
with $K^2 = \\Omega^2 \\mu \\epsilon$. Now we let
|
||||
with $K^2 = \Omega^2 \mu \epsilon$. Now we let
|
||||
|
||||
$$ \\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)} $$
|
||||
$$ \phi_{\vec{r}} = A e^{\imath (k_x m \Delta_x + k_y n \Delta_y + k_z p \Delta_z)} $$
|
||||
|
||||
resulting in
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
|
||||
\\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
|
||||
K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\
|
||||
\hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\
|
||||
K_x &= 2 \sin(k_x \Delta_x / 2) / \Delta_x \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
with similar expressions for the y and z dimnsions (and $K_y, K_z$).
|
||||
@ -485,20 +485,20 @@ with similar expressions for the y and z dimnsions (and $K_y, K_z$).
|
||||
This implies
|
||||
|
||||
$$
|
||||
\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
|
||||
K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
|
||||
\tilde{\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\
|
||||
K_x^2 + K_y^2 + K_z^2 = \Omega^2 \mu \epsilon = \Omega^2 / c^2
|
||||
$$
|
||||
|
||||
where $c = \\sqrt{\\mu \\epsilon}$.
|
||||
where $c = \sqrt{\mu \epsilon}$.
|
||||
|
||||
Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if
|
||||
Assuming real $(k_x, k_y, k_z), \omega$ will be real only if
|
||||
|
||||
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
|
||||
$$ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) $$
|
||||
|
||||
If $\\Delta_x = \\Delta_y = \\Delta_z$, this simplifies to $c \\Delta_t < \\Delta_x / \\sqrt{3}$.
|
||||
If $\Delta_x = \Delta_y = \Delta_z$, this simplifies to $c \Delta_t < \Delta_x / \sqrt{3}$.
|
||||
This last form can be interpreted as enforcing causality; the distance that light
|
||||
travels in one timestep (i.e., $c \\Delta_t$) must be less than the diagonal
|
||||
of the smallest cell ( $\\Delta_x / \\sqrt{3}$ when on a uniform cubic grid).
|
||||
travels in one timestep (i.e., $c \Delta_t$) must be less than the diagonal
|
||||
of the smallest cell ( $\Delta_x / \sqrt{3}$ when on a uniform cubic grid).
|
||||
|
||||
|
||||
Grid description
|
||||
@ -515,8 +515,8 @@ to make the illustration simpler; we need at least two cells in the x dimension
|
||||
demonstrate how nonuniform `dx` affects the various components.
|
||||
|
||||
Place the E fore-vectors at integer indices $r = (m, n, p)$ and the H back-vectors
|
||||
at fractional indices $r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},
|
||||
p + \\frac{1}{2})$. Remember that these are indices and not coordinates; they can
|
||||
at fractional indices $r + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2},
|
||||
p + \frac{1}{2})$. Remember that these are indices and not coordinates; they can
|
||||
correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.
|
||||
|
||||
Draw lines to denote the planes on which the H components and back-vectors are defined.
|
||||
@ -718,14 +718,14 @@ composed of the three diagonal tensor components:
|
||||
or
|
||||
|
||||
$$
|
||||
\\epsilon = \\begin{bmatrix} \\epsilon_{xx} & 0 & 0 \\\\
|
||||
0 & \\epsilon_{yy} & 0 \\\\
|
||||
0 & 0 & \\epsilon_{zz} \\end{bmatrix}
|
||||
\epsilon = \begin{bmatrix} \epsilon_{xx} & 0 & 0 \\
|
||||
0 & \epsilon_{yy} & 0 \\
|
||||
0 & 0 & \epsilon_{zz} \end{bmatrix}
|
||||
$$
|
||||
$$
|
||||
\\mu = \\begin{bmatrix} \\mu_{xx} & 0 & 0 \\\\
|
||||
0 & \\mu_{yy} & 0 \\\\
|
||||
0 & 0 & \\mu_{zz} \\end{bmatrix}
|
||||
\mu = \begin{bmatrix} \mu_{xx} & 0 & 0 \\
|
||||
0 & \mu_{yy} & 0 \\
|
||||
0 & 0 & \mu_{zz} \end{bmatrix}
|
||||
$$
|
||||
|
||||
where the off-diagonal terms (e.g. `epsilon_xy`) are assumed to be zero.
|
||||
|
||||
@ -62,7 +62,7 @@ def deriv_back(
|
||||
def curl_forward(
|
||||
dx_e: Sequence[NDArray[numpy.float_]] | None = None,
|
||||
) -> fdfield_updater_t:
|
||||
"""
|
||||
r"""
|
||||
Curl operator for use with the E field.
|
||||
|
||||
Args:
|
||||
@ -71,7 +71,7 @@ def curl_forward(
|
||||
|
||||
Returns:
|
||||
Function `f` for taking the discrete forward curl of a field,
|
||||
`f(E)` -> curlE $= \\nabla_f \\times E$
|
||||
`f(E)` -> curlE $= \nabla_f \times E$
|
||||
"""
|
||||
Dx, Dy, Dz = deriv_forward(dx_e)
|
||||
|
||||
@ -91,7 +91,7 @@ def curl_forward(
|
||||
def curl_back(
|
||||
dx_h: Sequence[NDArray[numpy.float_]] | None = None,
|
||||
) -> fdfield_updater_t:
|
||||
"""
|
||||
r"""
|
||||
Create a function which takes the backward curl of a field.
|
||||
|
||||
Args:
|
||||
@ -100,7 +100,7 @@ def curl_back(
|
||||
|
||||
Returns:
|
||||
Function `f` for taking the discrete backward curl of a field,
|
||||
`f(H)` -> curlH $= \\nabla_b \\times H$
|
||||
`f(H)` -> curlH $= \nabla_b \times H$
|
||||
"""
|
||||
Dx, Dy, Dz = deriv_back(dx_h)
|
||||
|
||||
|
||||
@ -1,4 +1,4 @@
|
||||
"""
|
||||
r"""
|
||||
Utilities for running finite-difference time-domain (FDTD) simulations
|
||||
|
||||
See the discussion of `Maxwell's Equations` in `meanas.fdmath` for basic
|
||||
@ -11,9 +11,9 @@ Timestep
|
||||
From the discussion of "Plane waves and the Dispersion relation" in `meanas.fdmath`,
|
||||
we have
|
||||
|
||||
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
|
||||
$$ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) $$
|
||||
|
||||
or, if $\\Delta_x = \\Delta_y = \\Delta_z$, then $c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}}$.
|
||||
or, if $\Delta_x = \Delta_y = \Delta_z$, then $c \Delta_t < \frac{\Delta_x}{\sqrt{3}}$.
|
||||
|
||||
Based on this, we can set
|
||||
|
||||
@ -27,81 +27,81 @@ Poynting Vector and Energy Conservation
|
||||
|
||||
Let
|
||||
|
||||
$$ \\begin{aligned}
|
||||
\\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
|
||||
&=& &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p} \\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p} \\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\
|
||||
& &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p} \\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\
|
||||
& &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1} \\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}})
|
||||
\\end{aligned}
|
||||
$$ \begin{aligned}
|
||||
\tilde{S}_{l, l', \vec{r}} &=& &\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}} \\
|
||||
&=& &\vec{x} (\tilde{E}^y_{l,m+1,n,p} \hat{H}^z_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^z_{l,m+1,n,p} \hat{H}^y_{l', \vec{r} + \frac{1}{2}}) \\
|
||||
& &+ &\vec{y} (\tilde{E}^z_{l,m,n+1,p} \hat{H}^x_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^x_{l,m,n+1,p} \hat{H}^z_{l', \vec{r} + \frac{1}{2}}) \\
|
||||
& &+ &\vec{z} (\tilde{E}^x_{l,m,n,p+1} \hat{H}^y_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^y_{l,m,n,p+1} \hat{H}^z_{l', \vec{r} + \frac{1}{2}})
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
where $\\vec{r} = (m, n, p)$ and $\\otimes$ is a modified cross product
|
||||
in which the $\\tilde{E}$ terms are shifted as indicated.
|
||||
where $\vec{r} = (m, n, p)$ and $\otimes$ is a modified cross product
|
||||
in which the $\tilde{E}$ terms are shifted as indicated.
|
||||
|
||||
By taking the divergence and rearranging terms, we can show that
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}}
|
||||
&= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}) \\\\
|
||||
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} -
|
||||
\\tilde{E}_{l, \\vec{r}} \\cdot \\hat{\\nabla} \\times \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
|
||||
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
|
||||
(-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
|
||||
\\hat{M}_{l, \\vec{r} + \\frac{1}{2}}) -
|
||||
\\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
|
||||
\\tilde{J}_{l', \\vec{r}}) \\\\
|
||||
&= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
|
||||
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
|
||||
- \\hat{H}_{l'} \\cdot \\hat{M}_{l} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\hat{\nabla} \cdot \tilde{S}_{l, l', \vec{r}}
|
||||
&= \hat{\nabla} \cdot (\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}}) \\
|
||||
&= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l, \vec{r}} -
|
||||
\tilde{E}_{l, \vec{r}} \cdot \hat{\nabla} \times \hat{H}_{l', \vec{r} + \frac{1}{2}} \\
|
||||
&= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot
|
||||
(-\tilde{\partial}_t \mu_{\vec{r} + \frac{1}{2}} \hat{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} -
|
||||
\hat{M}_{l, \vec{r} + \frac{1}{2}}) -
|
||||
\tilde{E}_{l, \vec{r}} \cdot (\hat{\partial}_t \tilde{\epsilon}_{\vec{r}} \tilde{E}_{l'+\frac{1}{2}, \vec{r}} +
|
||||
\tilde{J}_{l', \vec{r}}) \\
|
||||
&= \hat{H}_{l'} \cdot (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) -
|
||||
\tilde{E}_l \cdot (\epsilon / \Delta_t )(\tilde{E}_{l'+\frac{1}{2}} - \tilde{E}_{l'-\frac{1}{2}})
|
||||
- \hat{H}_{l'} \cdot \hat{M}_{l} - \tilde{E}_l \cdot \tilde{J}_{l'} \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
where in the last line the spatial subscripts have been dropped to emphasize
|
||||
the time subscripts $l, l'$, i.e.
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
|
||||
\\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}} \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\tilde{E}_l &= \tilde{E}_{l, \vec{r}} \\
|
||||
\hat{H}_l &= \tilde{H}_{l, \vec{r} + \frac{1}{2}} \\
|
||||
\tilde{\epsilon} &= \tilde{\epsilon}_{\vec{r}} \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
etc.
|
||||
For $l' = l + \\frac{1}{2}$ we get
|
||||
For $l' = l + \frac{1}{2}$ we get
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
|
||||
&= \\hat{H}_{l + \\frac{1}{2}} \\cdot
|
||||
(-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
|
||||
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} - \\tilde{E}_l)
|
||||
- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
|
||||
&= (-\\mu / \\Delta_t)(\\hat{H}^2_{l + \\frac{1}{2}} - \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}) -
|
||||
(\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} \\cdot \\tilde{E}_l - \\tilde{E}^2_l)
|
||||
- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
|
||||
&= -(\\mu \\hat{H}^2_{l + \\frac{1}{2}}
|
||||
+\\epsilon \\tilde{E}_{l+1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
|
||||
+(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
|
||||
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
|
||||
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}}
|
||||
&= \hat{H}_{l + \frac{1}{2}} \cdot
|
||||
(-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) -
|
||||
\tilde{E}_l \cdot (\epsilon / \Delta_t)(\tilde{E}_{l+1} - \tilde{E}_l)
|
||||
- \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\
|
||||
&= (-\mu / \Delta_t)(\hat{H}^2_{l + \frac{1}{2}} - \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}) -
|
||||
(\epsilon / \Delta_t)(\tilde{E}_{l+1} \cdot \tilde{E}_l - \tilde{E}^2_l)
|
||||
- \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\
|
||||
&= -(\mu \hat{H}^2_{l + \frac{1}{2}}
|
||||
+\epsilon \tilde{E}_{l+1} \cdot \tilde{E}_l) / \Delta_t \\
|
||||
+(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}
|
||||
+\epsilon \tilde{E}^2_l) / \Delta_t \\
|
||||
- \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
|
||||
- \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
and for $l' = l - \\frac{1}{2}$,
|
||||
and for $l' = l - \frac{1}{2}$,
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}
|
||||
&= (\\mu \\hat{H}^2_{l - \\frac{1}{2}}
|
||||
+\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
|
||||
-(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
|
||||
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
|
||||
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}}
|
||||
&= (\mu \hat{H}^2_{l - \frac{1}{2}}
|
||||
+\epsilon \tilde{E}_{l-1} \cdot \tilde{E}_l) / \Delta_t \\
|
||||
-(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}
|
||||
+\epsilon \tilde{E}^2_l) / \Delta_t \\
|
||||
- \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
|
||||
- \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
These two results form the discrete time-domain analogue to Poynting's theorem.
|
||||
@ -109,25 +109,25 @@ They hint at the expressions for the energy, which can be calculated at the same
|
||||
time-index as either the E or H field:
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
|
||||
U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
U_l &= \epsilon \tilde{E}^2_l + \mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} \\
|
||||
U_{l + \frac{1}{2}} &= \epsilon \tilde{E}_l \cdot \tilde{E}_{l + 1} + \mu \hat{H}^2_{l + \frac{1}{2}} \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
Rewriting the Poynting theorem in terms of the energy expressions,
|
||||
|
||||
$$
|
||||
\\begin{aligned}
|
||||
(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
|
||||
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
|
||||
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
|
||||
(U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
|
||||
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
|
||||
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
(U_{l+\frac{1}{2}} - U_l) / \Delta_t
|
||||
&= -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\
|
||||
- \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
|
||||
- \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
|
||||
(U_l - U_{l-\frac{1}{2}}) / \Delta_t
|
||||
&= -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\
|
||||
- \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
|
||||
- \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
This result is exact and should practically hold to within numerical precision. No time-
|
||||
@ -147,10 +147,10 @@ of the time-domain fields.
|
||||
The Ricker wavelet (normalized second derivative of a Gaussian) is commonly used for the pulse
|
||||
shape. It can be written
|
||||
|
||||
$$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$
|
||||
$$ f_r(t) = (1 - \frac{1}{2} (\omega (t - \tau))^2) e^{-(\frac{\omega (t - \tau)}{2})^2} $$
|
||||
|
||||
with $\\tau > \\frac{2 * \\pi}{\\omega}$ as a minimum delay to avoid a discontinuity at
|
||||
t=0 (assuming the source is off for t<0 this gives $\\sim 10^{-3}$ error at t=0).
|
||||
with $\tau > \frac{2 * \pi}{\omega}$ as a minimum delay to avoid a discontinuity at
|
||||
t=0 (assuming the source is off for t<0 this gives $\sim 10^{-3}$ error at t=0).
|
||||
|
||||
|
||||
|
||||
|
||||
@ -12,7 +12,7 @@ def poynting(
|
||||
h: fdfield_t,
|
||||
dxes: dx_lists_t | None = None,
|
||||
) -> fdfield_t:
|
||||
"""
|
||||
r"""
|
||||
Calculate the poynting vector `S` ($S$).
|
||||
|
||||
This is the energy transfer rate (amount of energy `U` per `dt` transferred
|
||||
@ -44,16 +44,16 @@ def poynting(
|
||||
|
||||
The full relationship is
|
||||
$$
|
||||
\\begin{aligned}
|
||||
(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
|
||||
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
|
||||
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
|
||||
(U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
|
||||
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
|
||||
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
|
||||
\\end{aligned}
|
||||
\begin{aligned}
|
||||
(U_{l+\frac{1}{2}} - U_l) / \Delta_t
|
||||
&= -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\
|
||||
- \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
|
||||
- \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
|
||||
(U_l - U_{l-\frac{1}{2}}) / \Delta_t
|
||||
&= -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\
|
||||
- \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
|
||||
- \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
These equalities are exact and should practically hold to within numerical precision.
|
||||
|
||||
Loading…
Reference in New Issue
Block a user