lots more fdmath documentation
This commit is contained in:
parent
163aa52420
commit
b58f8ebb65
@ -2,6 +2,8 @@
|
||||
|
||||
Basic discrete calculus for finite difference (fd) simulations.
|
||||
|
||||
TODO: short description of functional vs operator form
|
||||
|
||||
Discrete calculus
|
||||
=================
|
||||
|
||||
@ -10,37 +12,69 @@ This documentation and approach is roughly based on W.C. Chew's excellent
|
||||
which covers a superset of this material with similar notation and more detail.
|
||||
|
||||
|
||||
Derivatives
|
||||
-----------
|
||||
Derivatives and shifted values
|
||||
------------------------------
|
||||
|
||||
Define the discrete forward derivative as
|
||||
$$ [\\tilde{\\partial}_x f ]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
|
||||
or
|
||||
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]
|
||||
|
||||
Dx_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
|
||||
|
||||
Likewise, discrete reverse derivative is
|
||||
$$ [\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
|
||||
|
||||
or
|
||||
|
||||
Dx_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
|
||||
deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
|
||||
|
||||
The derivatives' arrays are shifted by a half-cell relative to the original function:
|
||||
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]
|
||||
_________________________
|
||||
| | | | |
|
||||
| f0 | f1 | f2 | f3 | function
|
||||
|_____|_____|_____|_____|
|
||||
| | | |
|
||||
| Df0 | Df1 | Df2 | Df3 forward derivative (periodic boundary)
|
||||
___|_____|_____|_____|____
|
||||
| | | |
|
||||
| Df1 | Df2 | Df3 | Df0 reverse derivative (periodic boundary)
|
||||
___|_____|_____|_____|____
|
||||
[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)
|
||||
__|_____|________|________|___
|
||||
|
||||
Periodic boundaries are used unless otherwise noted.
|
||||
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.
|
||||
|
||||
For the remainder of the `Discrete calculus` section, all figures will show
|
||||
constant-length cells in order to focus on the vector derivatives themselves.
|
||||
See the `Grid description` section below for additional information on this topic.
|
||||
|
||||
|
||||
Gradients and fore-vectors
|
||||
@ -222,10 +256,10 @@ Maxwell's Equations
|
||||
If we discretize both space (m,n,p) and time (l), Maxwell's equations become
|
||||
|
||||
$$ \\begin{align*}
|
||||
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=& -&\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
|
||||
&+& \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\hat{\\nabla} \\times \\hat{H}_{l,\\vec{r}} &=& &\\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
|
||||
&+& \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
||||
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
|
||||
+ \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\hat{\\nabla} \\times \\hat{H}_{l,\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
|
||||
+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
||||
\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
|
||||
\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
|
||||
\\end{align*} $$
|
||||
@ -238,31 +272,106 @@ If we discretize both space (m,n,p) and time (l), Maxwell's equations become
|
||||
\\end{align*} $$
|
||||
|
||||
where the spatial subscripts are abbreviated as \\( \\vec{r} = (m, n, p) \\) and
|
||||
\\( \\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}) \\).
|
||||
This is Yee's algorithm, written in a form analogous to Maxwell's equations.
|
||||
\\( \\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:
|
||||
|
||||
[figure: Yee cell]
|
||||
(m, n+1, p+1) _________________________ (m+1, n+1, p+1)
|
||||
/: /|
|
||||
/ : / |
|
||||
/ : / | Locations of the
|
||||
/ : / | E- and H-field components
|
||||
/ : / | for the E fore-vector at
|
||||
/ : / | r = (m, n, p) and its associated
|
||||
(m, n, p+1)/________________________/ | H back-vector at r + 1/2 =
|
||||
| : | | (m + 1/2, n + 1/2, p + 1/2)
|
||||
| : | | (the large cube's center)
|
||||
| Hx : | |
|
||||
| /: :.................|......| (m+1, n+1, p)
|
||||
|/ : / | /
|
||||
Ez..........Hy | /
|
||||
| Ey.......:..Hz | / This is the Yee discretization
|
||||
| / : / | / scheme ("Yee cell").
|
||||
| / : / | /
|
||||
|/ :/ | /
|
||||
r=(m, n, p)|___________Ex___________|/ (m+1, n, p)
|
||||
|
||||
|
||||
Each component forms its own grid, offset from the others:
|
||||
|
||||
[figure: E-fields for adjacent cells]
|
||||
________Ex(p+1, m+1)_____
|
||||
/: /|
|
||||
/ : / |
|
||||
/ : / |
|
||||
Ey(p+1) Ey(m+1, p+1)
|
||||
/ : / |
|
||||
/ Ez(n+1) / Ez(m+1, n+1)
|
||||
/__________Ex(p+1)_______/ |
|
||||
| : | |
|
||||
| : | | This figure shows which fore-vector
|
||||
| : | | each e-field component belongs to.
|
||||
| :.........Ex(n+1).|......| Indices are shortened; e.g. Ex(p+1)
|
||||
| / | / means "Ex for the fore-vector located
|
||||
Ez / Ez(m+1)/ at (m, n, p+1)".
|
||||
| Ey | /
|
||||
| / | Ey(m+1)
|
||||
| / | /
|
||||
|/ | /
|
||||
r=(m, n, p)|___________Ex___________|/
|
||||
|
||||
|
||||
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 $$
|
||||
implying that the discrete Maxwell's equations do not produce spurious charges.
|
||||
|
||||
TODO: Maxwell's equations explanation
|
||||
TODO: Maxwell's equations plaintext
|
||||
|
||||
Wave equation
|
||||
-------------
|
||||
|
||||
$$
|
||||
\\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} \\cdot \\tilde{E}_{l, \\vec{r}}
|
||||
= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}} $$
|
||||
Taking the backward curl of the \\( \\tilde{\\nabla} \\times \\tilde{E} \\) equation and
|
||||
replacing the resulting \\( \\hat{\\nabla} \\times \\hat{H} \\) term using its respective equation,
|
||||
and setting \\( \\hat{M} \\) to zero, we can form the discrete wave equation:
|
||||
|
||||
$$
|
||||
\\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}})
|
||||
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
|
||||
&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
|
||||
\\end{align*}
|
||||
$$
|
||||
|
||||
TODO: wave equation explanation
|
||||
TODO: wave equation plaintext
|
||||
|
||||
|
||||
Grid description
|
||||
================
|
||||
|
||||
The
|
||||
|
||||
TODO: explain dxes
|
||||
|
||||
"""
|
||||
|
||||
Loading…
Reference in New Issue
Block a user