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@ -14,22 +14,21 @@ Derivatives
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-----------
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-----------
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Define the discrete forward derivative as
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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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or
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Dx_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
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Dx_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
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or
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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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Likewise, discrete reverse 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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or
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Dx_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
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Dx_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
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or
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The derivatives' arrays are shifted by a half-cell relative to the original function:
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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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The derivatives are shifted by a half-cell relative to the original function:
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[figure: derivatives]
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_________________________
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_________________________
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| f0 | f1 | f2 | f3 | function
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| f0 | f1 | f2 | f3 | function
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@ -48,13 +47,30 @@ Gradients and fore-vectors
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--------------------------
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--------------------------
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Expanding to three dimensions, we can define two gradients
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Expanding to three dimensions, we can define two gradients
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$$ [\\tilde{\\nabla} f]_{n,m,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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$$ [\\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{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}} $$
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\\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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$$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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$$ [\\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{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}} $$
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\\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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or
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[code: gradients]
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grad_forward(f)[i,j,k] = [Dx_forward(f)[i, j, k],
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Dy_forward(f)[i, j, k],
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Dz_forward(f)[i, j, k]]
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= [(f[i + 1, j, k] - f[i, j, k]) / dx[i],
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(f[i, j + 1, k] - f[i, j, k]) / dy[i],
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(f[i, j, k + 1] - f[i, j, k]) / dz[i]]
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grad_back(f)[i,j,k] = [Dx_back(f)[i, j, k],
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Dy_back(f)[i, j, k],
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Dz_back(f)[i, j, k]]
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= [(f[i, j, k] - f[i - 1, j, k]) / dx[i],
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(f[i, j, k] - f[i, j - 1, k]) / dy[i],
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(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
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The three derivatives in the gradient cause shifts in different
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directions, so the x/y/z components of the resulting "vector" are defined
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directions, so the x/y/z components of the resulting "vector" are defined
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at different points: the x-component is shifted in the x-direction,
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at different points: the x-component is shifted in the x-direction,
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@ -70,18 +86,18 @@ on the direction of the shift. We write it as
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\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
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\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
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(m, n+1, p+1) _____________ (m+1, n+1, p+1)
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[figure: gradient / fore-vector]
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/: /|
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(m, n+1, p+1) ______________ (m+1, n+1, p+1)
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/ : / |
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/: /|
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/ : / |
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/ : / |
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(m, n, p+1)/____________/ | The derivatives are defined
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/ : / |
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| : | | at the Dx, Dy, Dz points,
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(m, n, p+1)/_____________/ | The forward derivatives are defined
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| :........|...| but the gradient fore-vector
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| : | | at the Dx, Dy, Dz points,
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Dz / | / is the set of all three
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| :.........|...| but the forward-gradient fore-vector
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| Dy | / and is said to be "located" at (m,n,p)
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Dz / | / is the set of all three
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| / | /
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| Dy | / and is said to be "located" at (m,n,p)
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(m, n, p)|/____Dx_____|/ (m+1, n, p)
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| / | /
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(m, n, p)|/_____Dx_____|/ (m+1, n, p)
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@ -100,23 +116,58 @@ There are also two divergences,
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[\\hat{\\partial}_y g^y]_{m,n,p} +
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[\\hat{\\partial}_y g^y]_{m,n,p} +
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[\\hat{\\partial}_z g^z]_{m,n,p} $$
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[\\hat{\\partial}_z g^z]_{m,n,p} $$
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or
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[code: divergences]
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div_forward(g)[i,j,k] = Dx_forward(gx)[i, j, k] +
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Dy_forward(gy)[i, j, k] +
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Dz_forward(gz)[i, j, k]
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= (gx[i + 1, j, k] - gx[i, j, k]) / dx[i] +
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(gy[i, j + 1, k] - gy[i, j, k]) / dy[i] +
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(gz[i, j, k + 1] - gz[i, j, k]) / dz[i]
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div_back(g)[i,j,k] = Dx_back(gx)[i, j, k] +
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Dy_back(gy)[i, j, k] +
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Dz_back(gz)[i, j, k]
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= (gx[i, j, k] - gx[i - 1, j, k]) / dx[i] +
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(gy[i, j, k] - gy[i, j - 1, k]) / dy[i] +
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(gz[i, j, k] - gz[i, j, k - 1]) / dz[i]
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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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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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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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\\( (m \\pm \\frac{1}{2},n,p) \\) etc.
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[figure: divergence]
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^^
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(m-1/2, n+1/2, p+1/2) _____||_______ (m+1/2, n+1/2, p+1/2)
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/: || ,, /|
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/ : || // / | The divergence at (m, n, p) (the center
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/ : // / | of this cube) of a fore-vector field
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(m-1/2, n-1/2, p+1/2)/_____________/ | is the sum of the outward-pointing
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| : | | fore-vector components, which are
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<==|== :.........|.====> located at the face centers.
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| / | /
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| / // | / Note that in a nonuniform grid, each
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| / // || | / dimension is normalized by the cell width.
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(m-1/2, n-1/2, p-1/2)|/___//_______|/ (m+1/2, n-1/2, p-1/2)
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'' ||
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VV
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Curls
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Curls
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-----
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-----
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The two curls are then
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The two curls are then
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$$ \\begin{align}
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$$ \\begin{align*}
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\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
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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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[\\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{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{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}_x g^z_{m + \\frac{1}{2},n,p})
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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} $$
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\\end{align*} $$
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and
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and
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@ -128,8 +179,90 @@ The two curls are then
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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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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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with components at \\((m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\) etc.
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TODO: draw diagrams for vector derivatives
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TODO: Explain fdfield_t vs vfdfield_t / operators vs functional
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[code: curls]
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curl_forward(g)[i,j,k] = [Dy_forward(gz)[i, j, k] - Dz_forward(gy)[i, j, k],
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Dz_forward(gx)[i, j, k] - Dx_forward(gz)[i, j, k],
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Dx_forward(gy)[i, j, k] - Dy_forward(gx)[i, j, k]]
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curl_back(g)[i,j,k] = [Dy_back(gz)[i, j, k] - Dz_back(gy)[i, j, k],
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Dz_back(gx)[i, j, k] - Dx_back(gz)[i, j, k],
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Dx_back(gy)[i, j, k] - Dy_back(gx)[i, j, k]]
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For example, consider the forward curl, at (m, n, p), of a back-vector field `g`, defined
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on a grid containing (m + 1/2, n + 1/2, p + 1/2).
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The curl will be a fore-vector, so its z-component will be defined at (m, n, p + 1/2).
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Take the nearest x- and y-components of `g` in the xy plane where the curl's z-component
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is located; these are
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[curl components]
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(m, n + 1/2, p + 1/2) : x-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
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(m + 1, n + 1/2, p + 1/2) : x-component of back-vector at (m + 3/2, n + 1/2, p + 1/2)
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(m + 1/2, n , p + 1/2) : y-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
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(m + 1/2, n + 1 , p + 1/2) : y-component of back-vector at (m + 1/2, n + 3/2, p + 1/2)
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These four xy-components can be used to form a loop around the curl's z-component; its magnitude and sign
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is set by their loop-oriented sum (i.e. two have their signs flipped to complete the loop).
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[figure: z-component of curl]
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: |
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: ^^ |
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:....||.<.....| (m, n+1, p+1/2)
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/ || /
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| v || | ^
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| / | /
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(m, n, p+1/2) |/_____>______|/ (m+1, n, p+1/2)
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Maxwell's Equations
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===================
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If we discretize both space (m,n,p) and time (l), Maxwell's equations become
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$$ \\begin{align*}
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\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=& -&\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
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&+& \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{l,\\vec{r}} &=& &\\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
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&+& \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
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\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
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\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
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\\end{align*} $$
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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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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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This is Yee's algorithm, written in a form analogous to Maxwell's equations.
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The divergence equations can be derived by taking the divergence of the curl equations
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and combining them with charge continuity,
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$$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$
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implying that the discrete Maxwell's equations do not produce spurious charges.
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TODO: Maxwell's equations explanation
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TODO: Maxwell's equations plaintext
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Wave equation
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-------------
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$$
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\\hat{\\nabla} \\times \\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}}
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+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
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= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}} $$
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TODO: wave equation explanation
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TODO: wave equation plaintext
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Grid description
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================
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TODO: explain dxes
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TODO: explain dxes
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"""
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"""
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