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add headings and vector diagram

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Jan Petykiewicz 2019-11-30 01:24:16 -08:00
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meanas/fdmath/__init__.py
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@ -1,11 +1,18 @@
""" """
Basic discrete calculus for finite difference (fd) simulations. Basic discrete calculus for finite difference (fd) simulations.
Discrete calculus
=================
This documentation and approach is roughly based on W.C. Chew's excellent This documentation and approach is roughly based on W.C. Chew's excellent
"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770), "Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770),
which covers a superset of this material with similar notation and more detail. which covers a superset of this material with similar notation and more detail.
Derivatives
-----------
Define the discrete forward derivative as Define the discrete forward derivative as
Dx_forward(f)[i] = (f[i + 1] - f[i]) / dx[i] Dx_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
@ -37,6 +44,9 @@ The derivatives are shifted by a half-cell relative to the original function:
Periodic boundaries are used unless otherwise noted. Periodic boundaries are used unless otherwise noted.
Gradients and fore-vectors
--------------------------
Expanding to three dimensions, we can define two gradients Expanding to three dimensions, we can define two gradients
$$ [\\tilde{\\nabla} f]_{n,m,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} + $$ [\\tilde{\\nabla} f]_{n,m,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
\\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} + \\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
@ -60,6 +70,24 @@ on the direction of the shift. We write it as
\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$ \\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
(m, n+1, p+1) _____________ (m+1, n+1, p+1)
/: /|
/ : / |
/ : / |
(m, n, p+1)/____________/ | The derivatives are defined
| : | | at the Dx, Dy, Dz points,
| :........|...| but the gradient fore-vector
Dz / | / is the set of all three
| Dy | / and is said to be "located" at (m,n,p)
| / | /
(m, n, p)|/____Dx_____|/ (m+1, n, p)
Divergences
-----------
There are also two divergences, There are also two divergences,
$$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p} $$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
@ -77,7 +105,11 @@ is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not
\\( (m \\pm \\frac{1}{2},n,p) \\) etc. \\( (m \\pm \\frac{1}{2},n,p) \\) etc.
Curls
-----
The two curls are then The two curls are then
$$ \\begin{align} $$ \\begin{align}
\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\ \\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}} &= [\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
@ -85,7 +117,9 @@ The two curls are then
&+ \\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{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}_x g^z_{m + \\frac{1}{2},n,p}) &+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_x g^z_{m + \\frac{1}{2},n,p})
\\end{align} $$ \\end{align} $$
and and
$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} = $$ \\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}} $$ [\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$