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103 lines
4.4 KiB
Python
103 lines
4.4 KiB
Python
"""
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Basic discrete calculus for finite difference (fd) simulations.
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This documentation and approach is roughly based on W.C. Chew's excellent
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"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770),
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which covers a superset of this material with similar notation and more detail.
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Define the discrete forward derivative as
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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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Dx_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
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or
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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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_________________________
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| | | | |
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| f0 | f1 | f2 | f3 |
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|_____|_____|_____|_____|
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| | | |
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| Df0 | Df1 | Df2 | Df3
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___|_____|_____|_____|____
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Periodic boundaries are used unless otherwise noted.
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Expanding to three dimensions, we can define two gradients
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$$ [\\tilde{\\nabla} f]_{n,m,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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\\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
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\\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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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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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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at different points: the x-component is shifted in the x-direction,
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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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There are also two divergences,
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$$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
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= [\\tilde{\\partial}_x g^x]_{m,n,p} +
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[\\tilde{\\partial}_y g^y]_{m,n,p} +
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[\\tilde{\\partial}_z g^z]_{m,n,p} $$
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$$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p}
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= [\\hat{\\partial}_x g^x]_{m,n,p} +
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[\\hat{\\partial}_y g^y]_{m,n,p} +
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[\\hat{\\partial}_z g^z]_{m,n,p} $$
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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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The two curls are then
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$$ \\begin{align}
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\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
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[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
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\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
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&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
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&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_x g^z_{m + \\frac{1}{2},n,p})
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\\end{align}$$
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and
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$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
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[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
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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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TODO: Explain fdfield_t vs vfdfield_t / operators vs functional
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TODO: explain dxes
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"""
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from .types import fdfield_t, vfdfield_t, dx_lists_t, fdfield_updater_t
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from .vectorization import vec, unvec
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from . import operators, functional, types, vectorization
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