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"""
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Basic discrete calculus for finite difference (fd) simulations.
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TODO: short description of functional vs operator form
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Discrete calculus
=================
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This documentation and approach is roughly based on W.C. Chew's excellent
"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770),
which covers a superset of this material with similar notation and more detail.
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Scalar derivatives and cell shifts
----------------------------------
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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 \\)).
The value at \\( m \\) occupies a length \\( \\Delta_{x, m} \\) along the x-axis. Note that \\( m \\)
is an index along the x-axis, _not_ necessarily an x-coordinate, since each length
\\( \\Delta_{x, m}, \\Delta_{x, m+1}, ...\\) is independently chosen.
If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]`
along the x-axis, the forward derivative is
deriv_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
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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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or
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deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
The derivatives' values are shifted by a half-cell relative to the original function, and
will have different cell widths if all the `dx[i]` ( \\( \\Delta_{x, m} \\) ) are not
identical:
[figure: derivatives and cell sizes]
dx0 dx1 dx2 dx3 cell sizes for function
----- ----- ----------- -----
______________________________
| | | |
f0 | f1 | f2 | f3 | function
_____|_____|___________|_____|
| | | |
| Df0 | Df1 | Df2 | Df3 forward derivative (periodic boundary)
__|_____|________|________|___
dx'3] dx'0 dx'1 dx'2 [dx'3 cell sizes for forward derivative
-- ----- -------- -------- ---
dx'0] dx'1 dx'2 dx'3 [dx'0 cell sizes for reverse derivative
______________________________
| | | |
| df1 | df2 | df3 | df0 reverse derivative (periodic boundary)
__|_____|________|________|___
Periodic boundaries are used here and elsewhere unless otherwise noted.
In the above figure,
`f0 =` \\(f_0\\), `f1 =` \\(f_1\\)
`Df0 =` \\([\\tilde{\\partial}f]_{0 + \\frac{1}{2}}\\)
`Df1 =` \\([\\tilde{\\partial}f]_{1 + \\frac{1}{2}}\\)
`df0 =` \\([\\hat{\\partial}f]_{0 - \\frac{1}{2}}\\)
etc.
The fractional subscript \\( m + \\frac{1}{2} \\) is used to indicate values defined
at shifted locations relative to the original \\( m \\), with corresponding lengths
$$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
Just as \\( m \\) is not itself an x-coordinate, neither is \\( m + \\frac{1}{2} \\);
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,
the positions labeled with \\( m + \\frac{1}{2} \\) are said to lie on a "dual" or
"derived" grid.
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For the remainder of the `Discrete calculus` section, all figures will show
constant-length cells in order to focus on the vector derivatives themselves.
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See the `Grid description` section below for additional information on this topic
and generalization to three dimensions.
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Gradients and fore-vectors
--------------------------
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Expanding to three dimensions, we can define two gradients
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$$ [\\tilde{\\nabla} f]_{m,n,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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\\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
\\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
$$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
\\vec{y} [\\hat{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
\\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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or
[code: gradients]
grad_forward(f)[i,j,k] = [Dx_forward(f)[i, j, k],
Dy_forward(f)[i, j, k],
Dz_forward(f)[i, j, k]]
= [(f[i + 1, j, k] - f[i, j, k]) / dx[i],
(f[i, j + 1, k] - f[i, j, k]) / dy[i],
(f[i, j, k + 1] - f[i, j, k]) / dz[i]]
grad_back(f)[i,j,k] = [Dx_back(f)[i, j, k],
Dy_back(f)[i, j, k],
Dz_back(f)[i, j, k]]
= [(f[i, j, k] - f[i - 1, j, k]) / dx[i],
(f[i, j, k] - f[i, j - 1, k]) / dy[i],
(f[i, j, k] - f[i, j, k - 1]) / dz[i]]
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The three derivatives in the gradient cause shifts in different
directions, so the x/y/z components of the resulting "vector" are defined
at different points: the x-component is shifted in the x-direction,
y in y, and z in z.
We call the resulting object a "fore-vector" or "back-vector", depending
on the direction of the shift. We write it as
$$ \\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p} +
\\vec{y} g^y_{m,n + \\frac{1}{2},p} +
\\vec{z} g^z_{m,n,p + \\frac{1}{2}} $$
$$ \\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} +
\\vec{y} g^y_{m,n - \\frac{1}{2},p} +
\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
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[figure: gradient / fore-vector]
(m, n+1, p+1) ______________ (m+1, n+1, p+1)
/: /|
/ : / |
/ : / |
(m, n, p+1)/_____________/ | The forward derivatives are defined
| : | | at the Dx, Dy, Dz points,
| :.........|...| but the forward-gradient fore-vector
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z y Dz / | / is the set of all three
|/_x | Dy | / and is said to be "located" at (m,n,p)
|/ |/
(m, n, p)|_____Dx______| (m+1, n, p)
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Divergences
-----------
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There are also two divergences,
$$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
= [\\tilde{\\partial}_x g^x]_{m,n,p} +
[\\tilde{\\partial}_y g^y]_{m,n,p} +
[\\tilde{\\partial}_z g^z]_{m,n,p} $$
$$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p}
= [\\hat{\\partial}_x g^x]_{m,n,p} +
[\\hat{\\partial}_y g^y]_{m,n,p} +
[\\hat{\\partial}_z g^z]_{m,n,p} $$
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or
[code: divergences]
div_forward(g)[i,j,k] = Dx_forward(gx)[i, j, k] +
Dy_forward(gy)[i, j, k] +
Dz_forward(gz)[i, j, k]
= (gx[i + 1, j, k] - gx[i, j, k]) / dx[i] +
(gy[i, j + 1, k] - gy[i, j, k]) / dy[i] +
(gz[i, j, k + 1] - gz[i, j, k]) / dz[i]
div_back(g)[i,j,k] = Dx_back(gx)[i, j, k] +
Dy_back(gy)[i, j, k] +
Dz_back(gz)[i, j, k]
= (gx[i, j, k] - gx[i - 1, j, k]) / dx[i] +
(gy[i, j, k] - gy[i, j - 1, k]) / dy[i] +
(gz[i, j, k] - gz[i, j, k - 1]) / dz[i]
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
is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not at the locations of its components
\\( (m \\pm \\frac{1}{2},n,p) \\) etc.
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[figure: divergence]
^^
(m-1/2, n+1/2, p+1/2) _____||_______ (m+1/2, n+1/2, p+1/2)
/: || ,, /|
/ : || // / | The divergence at (m, n, p) (the center
/ : // / | of this cube) of a fore-vector field
(m-1/2, n-1/2, p+1/2)/_____________/ | is the sum of the outward-pointing
| : | | fore-vector components, which are
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z y <==|== :.........|.====> located at the face centers.
|/_x | / | /
| / // | / Note that in a nonuniform grid, each
|/ // || |/ dimension is normalized by the cell width.
(m-1/2, n-1/2, p-1/2)|____//_______| (m+1/2, n-1/2, p-1/2)
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'' ||
VV
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Curls
-----
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The two curls are then
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$$ \\begin{align*}
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\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
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&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
\\end{align*} $$
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and
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$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
where \\( \\hat{g} \\) and \\( \\tilde{g} \\) are located at \\((m,n,p)\\)
with components at \\( (m \\pm \\frac{1}{2},n,p) \\) etc.,
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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]
curl_forward(g)[i,j,k] = [Dy_forward(gz)[i, j, k] - Dz_forward(gy)[i, j, k],
Dz_forward(gx)[i, j, k] - Dx_forward(gz)[i, j, k],
Dx_forward(gy)[i, j, k] - Dy_forward(gx)[i, j, k]]
curl_back(g)[i,j,k] = [Dy_back(gz)[i, j, k] - Dz_back(gy)[i, j, k],
Dz_back(gx)[i, j, k] - Dx_back(gz)[i, j, k],
Dx_back(gy)[i, j, k] - Dy_back(gx)[i, j, k]]
For example, consider the forward curl, at (m, n, p), of a back-vector field `g`, defined
on a grid containing (m + 1/2, n + 1/2, p + 1/2).
The curl will be a fore-vector, so its z-component will be defined at (m, n, p + 1/2).
Take the nearest x- and y-components of `g` in the xy plane where the curl's z-component
is located; these are
[curl components]
(m, n + 1/2, p + 1/2) : x-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
(m + 1, n + 1/2, p + 1/2) : x-component of back-vector at (m + 3/2, n + 1/2, p + 1/2)
(m + 1/2, n , p + 1/2) : y-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
(m + 1/2, n + 1 , p + 1/2) : y-component of back-vector at (m + 1/2, n + 3/2, p + 1/2)
These four xy-components can be used to form a loop around the curl's z-component; its magnitude and sign
is set by their loop-oriented sum (i.e. two have their signs flipped to complete the loop).
[figure: z-component of curl]
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: |
z y : ^^ |
|/_x :....||.<.....| (m, n+1, p+1/2)
/ || /
| v || | ^
|/ |/
(m, n, p+1/2) |_____>______| (m+1, n, p+1/2)
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Maxwell's Equations
===================
If we discretize both space (m,n,p) and time (l), Maxwell's equations become
$$ \\begin{align*}
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\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
+ \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times \\hat{H}_{l,\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
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\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
\\end{align*} $$
with
$$ \\begin{align*}
\\hat{B}_\\vec{r} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
\\tilde{D}_\\vec{r} &= \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r}
\\end{align*} $$
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}) \\),
\\( \\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:
[code: Maxwell's equations]
H[i, j, k] -= (curl_forward(E[t])[i, j, k] - M[t, i, j, k]) / mu[i, j, k]
E[i, j, k] += (curl_back( H[t])[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting
in distinct locations for all six E- and H-field components:
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[figure: Field components]
(m - 1/2,=> ____________Hx__________[H] <= r + 1/2 = (m + 1/2,
n + 1/2, /: /: /| n + 1/2,
z y p + 1/2) / : / : / | p + 1/2)
|/_x / : / : / |
/ : Ez__________Hy | Locations of the E- and
/ : : : /| | H-field components for the
(m - 1/2, / : : Ey...../.|..Hz [E] fore-vector at r = (m,n,p)
n - 1/2, =>/________________________/ | /| (the large cube's center)
p + 1/2) | : : / | | / | and [H] back-vector at r + 1/2
| : :/ | |/ | (the top right corner)
| : [E].......|.Ex |
| :.................|......| <= (m + 1/2, n + 1/2, p + 1/2)
| / | /
| / | /
| / | / This is the Yee discretization
| / | / scheme ("Yee cell").
r - 1/2 = | / | /
(m - 1/2, |/ |/
n - 1/2,=> |________________________| <= (m + 1/2, n - 1/2, p - 1/2)
p - 1/2)
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Each component forms its own grid, offset from the others:
[figure: E-fields for adjacent cells]
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H1__________Hx0_________H0
z y /: /|
|/_x / : / | This figure shows H back-vector locations
/ : / | H0, H1, etc. and their associated components
Hy1 : Hy0 | H0 = (Hx0, Hy0, Hz0) etc.
/ : / |
/ Hz1 / Hz0
H2___________Hx3_________H3 | The equivalent drawing for E would have
| : | | fore-vectors located at the cube's
| : | | center (and the centers of adjacent cubes),
| : | | with components on the cube's faces.
| H5..........Hx4...|......H4
| / | /
Hz2 / Hz2 /
| / | /
| Hy6 | Hy4
| / | /
|/ |/
H6__________Hx7__________H7
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The divergence equations can be derived by taking the divergence of the curl equations
and combining them with charge continuity,
$$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$
implying that the discrete Maxwell's equations do not produce spurious charges.
Wave equation
-------------
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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:
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$$
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\\begin{align*}
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
+ \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
\\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}) \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}})
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+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
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&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
\\end{align*}
$$
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Grid description
================
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As described in the section on scalar discrete derivatives above, cell widths
(`dx[i]`, `dy[j]`, `dz[k]`) along each axis can be arbitrary and independently
defined. Moreover, all field components are actually defined at "derived" or "dual"
positions, in-between the "base" grid points on one or more axes.
To get a better sense of how this works, let's start by drawing a grid with uniform
`dy` and `dz` and nonuniform `dx`. We will only draw one cell in the y and z dimensions
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
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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.
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Draw lines to denote the planes on which the H components and back-vectors are defined.
For simplicity, don't draw the equivalent planes for the E components and fore-vectors,
except as necessary to show their locations -- it's easiest to just connect them to their
grid figure fixup
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associated H-equivalents.
The result looks something like this:
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[figure: Component centers]
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p=
[H]__________Hx___________[H]_____Hx______[H] __ +1/2
z y /: /: /: /: /| | |
|/_x / : / : / : / : / | | |
/ : / : / : / : / | | |
Hy : Ez...........Hy : Ez......Hy | | |
/: : : : /: : : : /| | | |
/ : Hz : Ey....../.:..Hz : Ey./.|..Hz __ 0 | dz[0]
/ : /: : / / : /: : / / | /| | |
/_________________________/_______________/ | / | | |
| :/ : :/ | :/ : :/ | |/ | | |
| Ex : [E].......|..Ex : [E]..|..Ex | | |
| : | : | | | |
| [H]..........Hx....|......[H].....H|x.....[H] __ --------- (n=+1/2, p=-1/2)
| / | / | / / /
Hz / Hz / Hz / / /
| / | / | / / /
| Hy | Hy | Hy __ 0 / dy[0]
| / | / | / / /
| / | / | / / /
|/ |/ |/ / /
[H]__________Hx___________[H]_____Hx______[H] __ -1/2 /
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=n
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|------------|------------|-------|-------|
-1/2 0 +1/2 +1 +3/2 = m
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------------------------- ----------------
dx[0] dx[1]
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Part of a nonuniform "base grid", with labels specifying
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positions of the various field components. [E] fore-vectors
are at the cell centers, and [H] back-vectors are at the
vertices. H components along the near (-y) top (+z) edge
have been omitted to make the insides of the cubes easier
to visualize.
work on grid text
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This figure shows where all the components are located; however, it is also useful to show
what volumes those components are responsible for representing. Consider the Ex component:
two of its nearest neighbors are E fore-vectors, labeled `[E]` in the figure.
[figure: Ex volumes]
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p=
<_________________________________________> __ +1/2
z y << /: / /: >> | |
|/_x < < / : / / : > > | |
< < / : / / : > > | |
< < / : / / : > > | |
<: < / : : / : >: > | |
< : < / : : / : > : > __ 0 | dz[0]
< : < / : : / :> : > | |
<____________/____________________/_______> : > | |
< : < | : : | > : > | |
< Ex < | : Ex | > Ex > | |
< : < | : : | > : > | |
< : <....|.......:........:...|.......>...:...> __ --------- (n=+1/2, p=-1/2)
< : < | / : /| /> : > / /
< : < | / : / | / > : > / /
< :< | / :/ | / > :> / /
< < | / : | / > > _ 0 / dy[0]
< < | / | / > > / /
< < | / | / > > / /
<< |/ |/ >> / /
<____________|____________________|_______> __ -1/2 /
=n
|------------|------------|-------|-------|
-1/2 0 +1/2 +1 +3/2 = m
~------------ -------------------- -------~
dx'[-1] dx'[0] dx'[1]
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The Ex values are positioned on the x-faces of the base
grid. They represent the Ex field in volumes shifted by
a half-cell in the x-dimension, as shown here. Only the
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center cell (with width dx'[0]) is fully shown; the
other two are truncated (shown using >< markers).
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Note that the Ex positions are the in the same positions
as the previous figure; only the cell boundaries have moved.
Also note that the points at which Ex is defined are not
necessarily centered in the volumes they represent; non-
uniform cell sizes result in off-center volumes like the
center cell here.
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[figure: Hy volumes]
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p=
z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm __ +1/2 s
|/_x << m: m: >> | |
< < m : m : > > | | dz'[1]
< < m : m : > > | |
Hy........... m........Hy...........m......Hy > | |
< < m : m : > > | |
< ______ m_____:_______________m_____:_>______ __ 0
< < m /: m / > > | |
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm > | |
< < | / : | / > > | | dz'[0]
< < | / : | / > > | |
< < | / : | / > > | |
< wwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwwww __ s
< < |/ w |/ w> > / /
_____________|_____________________|________ > / /
< < | w | w > > / /
< Hy........|...w........Hy.......|...w...>..Hy _ 0 / dy[0]
< < | w | w > > / /
<< | w | w > > / /
< |w |w >> / /
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww __ -1/2 /
|------------|------------|--------|-------|
-1/2 0 +1/2 +1 +3/2 = m
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~------------ --------------------- -------~
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dx'[-1] dx'[0] dx'[1]
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The Hy values are positioned on the y-edges of the base
grid. Again here, the 'Hy' labels represent the same points
as in the basic grid figure above; the edges have shifted
by a half-cell along the x- and z-axes.
The grid lines _|:/ are edges of the area represented by
each Hy value, and the lines drawn using <m>.w represent
edges where a cell's faces extend beyond the drawn area
(i.e. where the drawing is truncated in the x- or z-
directions).
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lots more fdmath documentation
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move stuff under fdmath
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TODO: explain dxes
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[figure: 3D base and derived grids]
_____________________________ _____________________________
z y /: /: /: /| z y /: /: /:
|/_x / : / : / : / | |/_x / : / : / :
/ : / : / : / | / : / : / :
/___________________________/ | dz[1] ________________________/____
/ : / : / : /| | /: : / : /: : dz[1]
/: : / : / : / | | / : : / : / : :
/ : :..../......:/......:/..|...| / .:...:../......:/..:...:.....
/___________/_______/_______/ | /| ______/_________/_______/___: :
| : / : | | | | / | | : : | | : :
| : / : | | | |/ | | : : | | : :
| :/ : | | | | | dz[0] | : : | | : : dz[0]
| / : | | | /| | | : : | | : :
| /: :...|.......|.......|./ |...| | ..:...:.|.......|...:...:.....
|/ : / | /| /|/ | / | : / | /| : /
|___________|_______|_______| | / dy[1] ______|_________|_______|___: / dy[1]
| : / | / | / | |/ | :/ | / | :/
| :/.......|.../...|.../...|...| ..|...:.....|.../...|...:...
| / | / | / | / | / | / | / dy[0]
| / | / | / | / dy[0] | / | / | /
|/ |/ |/ |/ |/ |/ |/
|___________|_______|_______| ______|_________|_______|___
dx[0] dx[1] dx[2] dx'[0] dx'[1] dx'[2]
Base grid Shifted one half-cell right (e.g. for 1D
forward x derivative of all components).
z y : / : / :dz'[1] Some lines are omitted for clarity.
|/_x :/ :/ :/
.......:..........:.......:...
| /: | /: | /:
| / : | / : | / :
|/ : |/ : |/ :dz'[0]
______________________________
/| :/ /| :/ /| :/dy'[1]
/.|...:..../.|...:./.|...:....
| /: | /: | /:
| / : | / : | /dy'[0]
|/ : |/ : |/ :
_______________________________
/| /| /|
/ | / | / |
| | |
dx'[0] dx'[1] dx'[2]
All three dimensions shifted by one half-
cell. This is quite hard to visualize
(and probably not entirely to scale); see
later figures for a better representation.
documentation updates
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"""
move stuff under fdmath
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from .types import fdfield_t, vfdfield_t, dx_lists_t, fdfield_updater_t
from .vectorization import vec, unvec
from . import operators, functional, types, vectorization
Reference in New Issue Copy Permalink