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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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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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$$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
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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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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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@ -62,7 +62,7 @@ Likewise, discrete reverse derivative is
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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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@ -87,19 +87,20 @@ identical:
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Periodic boundaries are used here and elsewhere unless otherwise noted.
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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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`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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etc.
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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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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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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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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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"derived" grid.
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For the remainder of the `Discrete calculus` section, all figures will show
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@ -201,8 +202,8 @@ There are also two divergences,
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where `g = [gx, gy, gz]` is a fore- or back-vector field.
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Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
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is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not at the locations of its components
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\\( (m \\pm \\frac{1}{2},n,p) \\) etc.
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is defined at the back-vector's (fore-vector's) location $(m,n,p)$ and not at the locations of its components
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$(m \\pm \\frac{1}{2},n,p)$ etc.
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[figure: divergence]
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^^
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@ -226,23 +227,23 @@ Curls
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The two curls are then
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$$ \\begin{align*}
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$$ \\begin{aligned}
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\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
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[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
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\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
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&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
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&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
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\\end{align*} $$
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\\end{aligned} $$
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and
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$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
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[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
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where \\( \\hat{g} \\) and \\( \\tilde{g} \\) are located at \\((m,n,p)\\)
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with components at \\( (m \\pm \\frac{1}{2},n,p) \\) etc.,
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while \\( \\hat{h} \\) and \\( \\tilde{h} \\) are located at \\((m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\)
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with components at \\((m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\) etc.
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where $\\hat{g}$ and $\\tilde{g}$ are located at $(m,n,p)$
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with components at $(m \\pm \\frac{1}{2},n,p)$ etc.,
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while $\\hat{h}$ and $\\tilde{h}$ are located at $(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$
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with components at $(m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ etc.
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[code: curls]
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@ -286,27 +287,27 @@ Maxwell's Equations
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If we discretize both space (m,n,p) and time (l), Maxwell's equations become
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$$ \\begin{align*}
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$$ \\begin{aligned}
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\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
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- \\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
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+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
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\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
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\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
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\\end{align*} $$
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\\end{aligned} $$
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with
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$$ \\begin{align*}
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\\hat{B}_\\vec{r} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\tilde{D}_\\vec{r} &= \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r}
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\\end{align*} $$
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$$ \\begin{aligned}
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\\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}}
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\\end{aligned} $$
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where the spatial subscripts are abbreviated as \\( \\vec{r} = (m, n, p) \\) and
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\\( \\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}) \\),
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\\( \\tilde{E} \\) and \\( \\hat{H} \\) are the electric and magnetic fields,
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\\( \\tilde{J} \\) and \\( \\hat{M} \\) are the electric and magnetic current distributions,
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and \\( \\epsilon \\) and \\( \\mu \\) are the dielectric permittivity and magnetic permeability.
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where the spatial subscripts are abbreviated as $\\vec{r} = (m, n, p)$ and
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$\\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})$,
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$\\tilde{E}$ and $\\hat{H}$ are the electric and magnetic fields,
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$\\tilde{J}$ and $\\hat{M}$ are the electric and magnetic current distributions,
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and $\\epsilon$ and $\\mu$ are the dielectric permittivity and magnetic permeability.
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The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
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The time derivatives can be expanded to form the update equations:
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@ -375,12 +376,12 @@ and combining them with charge continuity,
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Wave equation
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-------------
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Taking the backward curl of the \\( \\tilde{\\nabla} \\times \\tilde{E} \\) equation and
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replacing the resulting \\( \\hat{\\nabla} \\times \\hat{H} \\) term using its respective equation,
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and setting \\( \\hat{M} \\) to zero, we can form the discrete wave equation:
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Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and
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replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation,
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and setting $\\hat{M}$ to zero, we can form the discrete wave equation:
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$$
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\\begin{align*}
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\\begin{aligned}
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\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
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-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
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- \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
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@ -391,11 +392,11 @@ $$
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
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-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
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-\\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}} \\\\
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-\\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}} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
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+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
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+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
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&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
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\\end{align*}
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\\end{aligned}
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$$
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@ -405,27 +406,27 @@ Frequency domain
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We can substitute in a time-harmonic fields
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$$
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\\begin{align*}
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\\tilde{E}_\\vec{r} &= \\tilde{E}_\\vec{r} e^{-\\imath \\omega l \\Delta_t} \\\\
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\\tilde{J}_\\vec{r} &= \\tilde{J}_\\vec{r} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
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\\end{align*}
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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{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{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{align*}
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\\begin{aligned}
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\\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}\\\\
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\\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}\\\\
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\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
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\\end{align*}
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\\end{aligned}
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$$
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This gives the frequency-domain wave equation,
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$$
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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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$$
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@ -435,69 +436,69 @@ Plane waves and Dispersion relation
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With uniform material distribution and no sources
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$$
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\\begin{align*}
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\\begin{aligned}
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\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
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\\epsilon_\\vec{r} &= \\epsilon \\\\
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\\tilde{J}_\\vec{r} &= 0 \\\\
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\\end{align*}
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\\epsilon_{\\vec{r}} &= \\epsilon \\\\
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\\tilde{J}_{\\vec{r}} &= 0 \\\\
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\\end{aligned}
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$$
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the frequency domain wave equation simplifies to
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$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r} - \\Omega^2 \\epsilon \\mu \\tilde{E}_\\vec{r} = 0 $$
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$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
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Since \\( \\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r} = 0 \\), we can simplify
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Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify
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$$
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\\begin{align*}
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\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r}
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&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
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&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
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&= - \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r}
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\\end{align*}
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\\begin{aligned}
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\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
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&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
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&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
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&= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
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\\end{aligned}
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$$
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and we get
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$$ \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r} + \\Omega^2 \\epsilon \\mu \\tilde{E}_\\vec{r} = 0 $$
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$$ \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} + \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
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We can convert this to three scalar-wave equations of the form
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$$ (\\tilde{\\nabla}^2 + K^2) \\phi_\\vec{r} = 0 $$
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$$ (\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0 $$
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with \\( K^2 = \\Omega^2 \\mu \\epsilon \\). Now we let
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with $K^2 = \\Omega^2 \\mu \\epsilon$. Now we let
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$$ \\phi_\\vec{r} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)} $$
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$$ \\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)} $$
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resulting in
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$$
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\\begin{align*}
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\\begin{aligned}
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\\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}\\\\
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\\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}\\\\
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K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
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\\end{align*}
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\\end{aligned}
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$$
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with similar expressions for the y and z dimnsions (and \\( K_y, K_z \\)).
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with similar expressions for the y and z dimnsions (and $K_y, K_z$).
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This implies
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$$
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\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_\\vec{r} \\\\
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\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
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K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
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$$
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where \\( c = \\sqrt{\\mu \\epsilon} \\).
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where $c = \\sqrt{\\mu \\epsilon}$.
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Assuming real \\( (k_x, k_y, k_z), \\omega \\) will be real only if
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Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if
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$$ 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}) $$
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If \\( \\Delta_x = \\Delta_y = \\Delta_z \\), this simplifies to \\( c \\Delta_t < \\Delta_x / \\sqrt{3} \\).
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If $\\Delta_x = \\Delta_y = \\Delta_z$, this simplifies to $c \\Delta_t < \\Delta_x / \\sqrt{3}$.
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This last form can be interpreted as enforcing causality; the distance that light
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travels in one timestep (i.e., \\( c \\Delta_t \\)) must be less than the diagonal
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of the smallest cell ( \\( \\Delta_x / \\sqrt{3} \\) when on a uniform cubic grid).
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travels in one timestep (i.e., $c \\Delta_t$) must be less than the diagonal
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of the smallest cell ( $\\Delta_x / \\sqrt{3}$ when on a uniform cubic grid).
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Grid description
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@ -513,9 +514,9 @@ To get a better sense of how this works, let's start by drawing a grid with unif
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to make the illustration simpler; we need at least two cells in the x dimension to
|
||||
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
|
||||
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
|
||||
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.
|
||||
|
|
|
|||
|
|
@ -63,7 +63,7 @@ def curl_forward(dx_e: List[numpy.ndarray] = None) -> fdfield_updater_t:
|
|||
|
||||
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)
|
||||
|
||||
|
|
@ -90,7 +90,7 @@ def curl_back(dx_h: List[numpy.ndarray] = None) -> fdfield_updater_t:
|
|||
|
||||
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)
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue