[docs] correct Yee calculus, source polarity, and phasor conventions

meanas/fdfd/__init__.py

@@ -19,38 +19,39 @@ Submodules:
 
 ================================================================
 
-From the "Frequency domain" section of `meanas.fdmath`, we have
+Following the "Frequency domain" section of `meanas.fdmath`, we use
+$e^{+\imath \omega t}$ time dependence. Phasors are referenced to the
+actual sample times of the staggered fields and sources:
 
 $$
  \begin{aligned}
- \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
+ \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{+\imath \omega l \Delta_t} \\
- \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
+ \hat{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \hat{H}_{\vec{r} + \frac{1}{2}} e^{+\imath \omega (l - \frac{1}{2}) \Delta_t} \\
- \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
+ \tilde{J}_{l - \frac{1}{2}, \vec{r}} &= \tilde{J}_{\vec{r}} e^{+\imath \omega (l - \frac{1}{2}) \Delta_t} \\
- \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\
+ \hat{M}_{l, \vec{r} + \frac{1}{2}} &= \hat{M}_{\vec{r} + \frac{1}{2}} e^{+\imath \omega l \Delta_t} \\
- \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
-    -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
  \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
  \end{aligned}
 $$
 
-resulting in
+The difference multipliers for an integer-indexed sequence are
 
 $$
  \begin{aligned}
- \tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
+ \tilde{\partial}_t &\Rightarrow \imath \Omega e^{ \imath \omega \Delta_t / 2}\\
-   \hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
+   \hat{\partial}_t &\Rightarrow \imath \Omega e^{-\imath \omega \Delta_t / 2}\\
  \end{aligned}
 $$
 
-Maxwell's equations are then
+The half-step factors cancel when the staggered sample times are included.
+Maxwell's equations for the phasors are then
 
 $$
   \begin{aligned}
   \tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
-         \imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}}
+        -\imath \Omega \hat{B}_{\vec{r} + \frac{1}{2}}
                                                      - \hat{M}_{\vec{r} + \frac{1}{2}}  \\
   \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
-        -\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}}
+         \imath \Omega \tilde{D}_{\vec{r}}
                                                      + \tilde{J}_{\vec{r}} \\
   \tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\
   \hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}}
@@ -61,13 +62,9 @@ With $\Delta_t \to 0$, this simplifies to
 
 $$
  \begin{aligned}
- \tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\
- \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\
- \tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\
- \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\
  \Omega &\to \omega \\
- \tilde{\partial}_t &\to -\imath \omega \\
+ \tilde{\partial}_t &\to \imath \omega \\
-   \hat{\partial}_t &\to -\imath \omega \\
+   \hat{\partial}_t &\to \imath \omega \\
  \end{aligned}
 $$
 
@@ -76,14 +73,17 @@ and then
 $$
   \begin{aligned}
   \tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
-         \imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
+        -\imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
                      - \hat{M}_{\vec{r} + \frac{1}{2}}  \\
   \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
-        -\imath \omega \tilde{D}_{\vec{r}}
+         \imath \omega \tilde{D}_{\vec{r}}
                      + \tilde{J}_{\vec{r}} \\
  \end{aligned}
 $$
 
+For zero magnetic current, eliminating $H$ gives the electric wave equation
+used by `operators.e_full` and `functional.e_full`:
+
 $$
  \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
     -\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\

meanas/fdmath/__init__.py

@@ -116,9 +116,9 @@ Expanding to three dimensions, we can define two gradients
   $$ [\tilde{\nabla} f]_{m,n,p} = \vec{x} [\tilde{\partial}_x f]_{m + \frac{1}{2},n,p} +
                                   \vec{y} [\tilde{\partial}_y f]_{m,n + \frac{1}{2},p} +
                                   \vec{z} [\tilde{\partial}_z f]_{m,n,p + \frac{1}{2}}  $$
-  $$ [\hat{\nabla} f]_{m,n,p} = \vec{x} [\hat{\partial}_x f]_{m + \frac{1}{2},n,p} +
+  $$ [\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{y} [\hat{\partial}_y f]_{m,n - \frac{1}{2},p} +
-                                \vec{z} [\hat{\partial}_z f]_{m,n,p + \frac{1}{2}}  $$
+                                \vec{z} [\hat{\partial}_z f]_{m,n,p - \frac{1}{2}}  $$
 
  or
 
@@ -232,7 +232,7 @@ The two curls are then
      [\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}}) \\
-     &+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^z_{m + \frac{1}{2},n,p})
+     &+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^x_{m + \frac{1}{2},n,p})
      \end{aligned} $$
 
  and
@@ -314,7 +314,7 @@ The time derivatives can be expanded to form the update equations:
 
     [code: Maxwell's equations updates]
     H[i, j, k] -= dt * (curl_forward(E)[i, j, k] + M[t, i, j, k]) /      mu[i, j, k]
-    E[i, j, k] += dt * (curl_back(   H)[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
+    E[i, j, k] += dt * (curl_back(   H)[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:
@@ -384,7 +384,7 @@ $$
   \begin{aligned}
   \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
      -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}
-                       - \hat{M}_{l-1, \vec{r} + \frac{1}{2}}  \\
+                       - \hat{M}_{l, \vec{r} + \frac{1}{2}}  \\
   \mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
    -\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}  \\
   \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
@@ -392,10 +392,10 @@ $$
   \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}} \\
+   -\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} - \tilde{\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}}
+           &= -\tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}}
   \end{aligned}
 $$
 
@@ -403,37 +403,42 @@ $$
 Frequency domain
 ----------------
 
-We can substitute in a time-harmonic fields
+We use the $e^{+\imath \omega t}$ convention, matching the FDFD operators.
+Substitute time-harmonic fields with phases referenced to their actual sample times:
 
 $$
  \begin{aligned}
- \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
+ \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{+\imath \omega l \Delta_t} \\
- \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t}
+ \tilde{J}_{l-\frac{1}{2}, \vec{r}} &= \tilde{J}_{\vec{r}} e^{+\imath \omega (l - \frac{1}{2}) \Delta_t}
  \end{aligned}
 $$
 
-resulting in
+For an integer-indexed sequence proportional to $e^{+\imath \omega l \Delta_t}$,
+the forward and backward difference multipliers are
 
 $$
  \begin{aligned}
- \tilde{\partial}_t &\Rightarrow (e^{ \imath \omega \Delta_t} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{-\imath \omega \Delta_t / 2} = -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
+ \tilde{\partial}_t &\Rightarrow (e^{ \imath \omega \Delta_t} - 1) / \Delta_t = \frac{2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{ \imath \omega \Delta_t / 2} = \imath \Omega e^{ \imath \omega \Delta_t / 2}\\
-   \hat{\partial}_t &\Rightarrow (1 - e^{-\imath \omega \Delta_t}) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{ \imath \omega \Delta_t / 2} = -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
+   \hat{\partial}_t &\Rightarrow (1 - e^{-\imath \omega \Delta_t}) / \Delta_t = \frac{2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{-\imath \omega \Delta_t / 2} = \imath \Omega e^{-\imath \omega \Delta_t / 2}\\
  \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
  \end{aligned}
 $$
 
-This gives the frequency-domain wave equation,
+The half-step factors account for the shift in the derivative's sample time.
+In particular, $\tilde{\partial}_t J_{l-\frac{1}{2}} = \imath \Omega J e^{\imath \omega l \Delta_t}$.
+Thus, when the staggered fields are expressed using the phasors above, the
+frequency-domain wave equation has no extra half-step phase:
 
 $$
  \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
-    -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
+    -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} \\
 $$
 
 
 Plane waves and Dispersion relation
 ------------------------------------
 
-With uniform material distribution and no sources
+With uniform grid spacings, a uniform material distribution, and no sources
 
 $$
  \begin{aligned}
@@ -474,31 +479,31 @@ resulting in
 
 $$
  \begin{aligned}
- \tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\
+ \tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_x = \frac{2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\
-   \hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\
+   \hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_x = \frac{2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\
  K_x &= 2 \sin(k_x \Delta_x / 2) / \Delta_x \\
  \end{aligned}
 $$
 
-with similar expressions for the y and z dimnsions (and $K_y, K_z$).
+with similar expressions for the y and z dimensions (and $K_y, K_z$).
 
 This implies
 
 $$
-  \tilde{\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\
+  \tilde{\nabla}^2 \phi_{\vec{r}} = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\
   K_x^2 + K_y^2 + K_z^2 = \Omega^2 \mu \epsilon = \Omega^2 / c^2
 $$
 
-where $c = \sqrt{\mu \epsilon}$.
+where $c = 1 / \sqrt{\mu \epsilon}$.
 
-Assuming real $(k_x, k_y, k_z), \omega$ will be real only if
+To keep $\omega$ real for every real grid wavevector $(k_x, k_y, k_z)$,
+we use the conservative stability bound
 
 $$ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) $$
 
 If $\Delta_x = \Delta_y = \Delta_z$, this simplifies to $c \Delta_t < \Delta_x / \sqrt{3}$.
-This last form can be interpreted as enforcing causality; the distance that light
+This limits the distance that light travels in one timestep to less than
-travels in one timestep (i.e., $c \Delta_t$) must be less than the diagonal
+$\Delta_x / \sqrt{3}$ on a uniform cubic grid.
-of the smallest cell ( $\Delta_x / \sqrt{3}$ when on a uniform cubic grid).
 
 
 Grid description

meanas/fdmath/functional.py

@@ -3,7 +3,6 @@ Math functions for finite difference simulations
 
 Basic discrete calculus etc.
 """
-from typing import TypeVar
 from collections.abc import Sequence, Callable
 
 import numpy
@@ -32,14 +31,15 @@ def _derivative(axis: int, forward: bool, widths: NDArray | None) -> fdfield_upd
                 raise ValueError('out must not alias the field or widths')
             if not output.flags.writeable or not numpy.can_cast(dtype, output.dtype, casting='same_kind'):
                 raise ValueError('out must be writable with a compatible dtype')
+            dtype = numpy.result_type(dtype, output.dtype)
         low, high = [slice(None)] * 3, [slice(None)] * 3
         low[axis], high[axis] = slice(None, -1), slice(1, None)
         first, last = [slice(None)] * 3, [slice(None)] * 3
         first[axis], last[axis] = slice(0, 1), slice(-1, None)
         destination = low if forward else high
-        numpy.subtract(f[tuple(high)], f[tuple(low)], out=output[tuple(destination)])
+        numpy.subtract(f[tuple(high)], f[tuple(low)], out=output[tuple(destination)], dtype=dtype)
         destination = last if forward else first
-        numpy.subtract(f[tuple(first)], f[tuple(last)], out=output[tuple(destination)])
+        numpy.subtract(f[tuple(first)], f[tuple(last)], out=output[tuple(destination)], dtype=dtype)
         if widths is not None:
             expand = [1, 1, 1]
             expand[axis] = len(widths)
@@ -55,6 +55,7 @@ def deriv_forward(
 
     Each callable accepts ``derivative(field, *, out=None)``. An output buffer
     must match the field shape and must not alias the field or width arrays.
+    Subtraction uses the dtype promoted by the field, widths, and output buffer.
     Slice-based subtraction avoids a full-size roll temporary.
     """
     return tuple(_derivative(axis, True, None if dx_e is None else dx_e[axis]) for axis in range(3))
@@ -70,12 +71,9 @@ def deriv_back(
     return tuple(_derivative(axis, False, None if dx_h is None else dx_h[axis]) for axis in range(3))
 
 
-TT = TypeVar('TT', bound='NDArray[floating | complexfloating]')
-
-
 def curl_forward(
         dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
-        ) -> Callable[[TT], TT]:
+        ) -> Callable[[NDArray[floating | complexfloating]], NDArray[floating | complexfloating]]:
     r"""
     Curl operator for use with the E field.
 
@@ -85,12 +83,14 @@ def curl_forward(
 
     Returns:
         Function `f` for taking the discrete forward curl of a field,
-        `f(E)` -> curlE $= \nabla_f \times E$
+        `f(E)` -> curlE $= \nabla_f \times E$.
+        The output dtype is promoted from the field and cell widths.
     """
     Dx, Dy, Dz = deriv_forward(dx_e)
 
-    def ce_fun(e: TT) -> TT:
+    def ce_fun(e: NDArray[floating | complexfloating]) -> NDArray[floating | complexfloating]:
-        output = numpy.empty_like(e)
+        dtype = e.dtype if dx_e is None else numpy.result_type(e, *dx_e)
+        output = numpy.empty_like(e, dtype=dtype)
         output[0] = Dy(e[2])
         output[1] = Dz(e[0])
         output[2] = Dx(e[1])
@@ -104,7 +104,7 @@ def curl_forward(
 
 def curl_back(
         dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
-        ) -> Callable[[TT], TT]:
+        ) -> Callable[[NDArray[floating | complexfloating]], NDArray[floating | complexfloating]]:
     r"""
     Create a function which takes the backward curl of a field.
 
@@ -114,12 +114,14 @@ def curl_back(
 
     Returns:
         Function `f` for taking the discrete backward curl of a field,
-        `f(H)` -> curlH $= \nabla_b \times H$
+        `f(H)` -> curlH $= \nabla_b \times H$.
+        The output dtype is promoted from the field and cell widths.
     """
     Dx, Dy, Dz = deriv_back(dx_h)
 
-    def ch_fun(h: TT) -> TT:
+    def ch_fun(h: NDArray[floating | complexfloating]) -> NDArray[floating | complexfloating]:
-        output = numpy.empty_like(h)
+        dtype = h.dtype if dx_h is None else numpy.result_type(h, *dx_h)
+        output = numpy.empty_like(h, dtype=dtype)
         output[0] = Dy(h[2])
         output[1] = Dz(h[0])
         output[2] = Dx(h[1])

meanas/fdmath/operators.py

@@ -61,6 +61,9 @@ def shift_with_mirror(
     Utility operator for performing an n-element shift along a specified axis, with mirror
     boundary conditions applied to the cells beyond the receding edge.
 
+    The absolute shift distance may not exceed the axis length. A shift by the
+    full axis length reverses that axis; shifts on a singleton axis are identities.
+
     Args:
         axis: Axis to shift along. x=0, y=1, z=2
         shape: Shape of the grid being shifted
@@ -73,8 +76,8 @@ def shift_with_mirror(
         raise Exception(f'Invalid shape: {shape}')
     if axis not in range(len(shape)):
         raise Exception(f'Invalid direction: {axis}, shape is {shape}')
-    if shift_distance >= shape[axis]:
+    if abs(shift_distance) > shape[axis]:
-        raise Exception(f'Shift ({shift_distance}) is too large for axis {axis} of size {shape[axis]}')
+        raise ValueError(f'Shift ({shift_distance}) is too large for axis {axis} of size {shape[axis]}')
 
     def mirrored_range(n: int, s: int) -> NDArray[numpy.int_]:
         v = numpy.arange(n) + s

meanas/fdmath/vectorization.py

@@ -113,7 +113,7 @@ def unvec(
     Returns `None` if called with `v=None`.
 
     Args:
-        v: 1D ndarray representing a vector field of shape shape (or None)
+        v: 1D array-like representing a vector field of shape shape (or None)
         shape: shape of the vector field
         nvdim: Number of components in each vector
 
@@ -122,4 +122,4 @@ def unvec(
     """
     if v is None:
         return None
-    return v.reshape((nvdim, *shape), order='C')  # type: ignore
+    return numpy.asarray(v).reshape((nvdim, *shape), order='C')

meanas/fdtd/base.py

@@ -17,13 +17,13 @@ def maxwell_e(
     """
     Build a function which performs a portion the time-domain E-field update,
 
-        E += curl_back(H[t]) / epsilon
+        E += dt * curl_back(H[t]) / epsilon
 
     The full update should be
 
-        E += (curl_back(H[t]) + J) / epsilon
+        E += dt * (curl_back(H[t]) - J) / epsilon
 
-    which requires an additional step of `E += J / epsilon` which is not performed
+    which requires an additional step of `E -= dt * J / epsilon` which is not performed
     by the generated function.
 
     See `meanas.fdmath` for descriptions of
@@ -72,13 +72,13 @@ def maxwell_h(
     """
     Build a function which performs part of the time-domain H-field update,
 
-        H -= curl_forward(E[t]) / mu
+        H -= dt * curl_forward(E[t]) / mu
 
     The full update should be
 
-        H -= (curl_forward(E[t]) + M) / mu
+        H -= dt * (curl_forward(E[t]) + M) / mu
 
-    which requires an additional step of `H -= M / mu` which is not performed
+    which requires an additional step of `H -= dt * M / mu` which is not performed
     by the generated function; this step can be omitted if there is no magnetic
     current `M`.
 

meanas/test/test_fdmath_functional.py

@@ -62,6 +62,55 @@ def test_curls_match_sparse_operators_on_nonuniform_grid() -> None:
     assert_fields_close(curl_back, matrix_back, atol=1e-12, rtol=1e-12)
 
 
+@pytest.mark.parametrize('name', ['curl_forward', 'curl_back'])
+@pytest.mark.parametrize('field_dtype', [numpy.float32, numpy.complex64])
+@pytest.mark.parametrize('complex_widths', [False, True])
+def test_curls_promote_field_dtype_and_match_sparse_operators(name, field_dtype, complex_widths) -> None:
+    field = VECTOR_FIELD.astype(field_dtype) if numpy.issubdtype(field_dtype, numpy.complexfloating) else VECTOR_FIELD.real.astype(field_dtype)
+    widths = [dx * (1 + 0.5j) if complex_widths else dx for dx in DX_E]
+    result = getattr(fd_functional, name)(widths)(field)
+    expected = unvec(getattr(fd_operators, name)(widths) @ vec(field), SHAPE)
+    assert result.dtype == numpy.result_type(field, *widths)
+    assert_fields_close(result, expected, atol=1e-12, rtol=1e-12)
+
+
+@pytest.mark.parametrize('name', ['curl_forward', 'curl_back'])
+def test_curls_preserve_small_residual_between_large_derivative_terms(name) -> None:
+    shape = (1, 2, 2)
+    field = numpy.zeros((3, *shape), dtype=numpy.float32)
+    field[2, :, 1, :] = 1e8
+    field[1, :, :, 1] = 1e8
+    widths = [numpy.ones(1), numpy.full(2, 1e8 / (1e8 + 1)), numpy.ones(2)]
+    result = getattr(fd_functional, name)(widths)(field)
+    expected = unvec(getattr(fd_operators, name)(widths) @ vec(field), shape)
+    assert abs(expected[0, 0, 0, 0]) == pytest.approx(1)
+    assert_fields_close(result, expected, atol=1e-7, rtol=1e-12)
+
+
+@pytest.mark.parametrize('name', ['deriv_forward', 'deriv_back'])
+@pytest.mark.parametrize('axis', [0, 1, 2])
+@pytest.mark.parametrize('spaced', [False, True])
+def test_derivatives_promote_subtraction_before_overflow(name, axis, spaced) -> None:
+    shape = [1, 1, 1]
+    shape[axis] = 2
+    field = numpy.array([3e38, -3e38], dtype=numpy.float32).reshape(shape)
+    widths = [numpy.full(n, 1e38) for n in shape] if spaced else None
+    derivative = getattr(fd_functional, name)(widths)[axis]
+    wide_field = field.astype(numpy.float64)
+    if name == 'deriv_forward':
+        expected = numpy.roll(wide_field, -1, axis=axis) - wide_field
+    else:
+        expected = wide_field - numpy.roll(wide_field, 1, axis=axis)
+    if spaced:
+        expected /= 1e38
+    with numpy.errstate(over='raise', invalid='raise'):
+        if spaced:
+            assert_close(derivative(field), expected)
+        output = numpy.empty(shape, dtype=numpy.float64)
+        assert derivative(field, out=output) is output
+        assert_close(output, expected)
+
+
 @pytest.mark.parametrize('periodic', [True, False])
 def test_gridlock_spacings_differentiate_physical_coordinates(periodic) -> None:
     grid = Grid([[0, 1, 3, 7, 15], [-2, 0, 5, 6], [3, 6, 10]], periodic=periodic)

meanas/test/test_fdmath_operators.py

@@ -56,7 +56,8 @@ def test_shift_circ_rejects_invalid_arguments(args: tuple[int, tuple[int, ...],
     [
         ((0, (2,), 1), 'Invalid shape'),
         ((3, SHAPE, 1), 'Invalid direction'),
-        ((0, SHAPE, SHAPE[0]), 'too large'),
+        ((0, SHAPE, SHAPE[0] + 1), 'too large'),
+        ((0, SHAPE, -SHAPE[0] - 1), 'too large'),
     ],
     )
 def test_shift_with_mirror_rejects_invalid_arguments(args: tuple[int, tuple[int, ...], int], message: str) -> None:
@@ -64,6 +65,23 @@ def test_shift_with_mirror_rejects_invalid_arguments(args: tuple[int, tuple[int,
         operators.shift_with_mirror(*args)
 
 
+@pytest.mark.parametrize('axis', [0, 1, 2])
+@pytest.mark.parametrize('polarity', [-1, 1])
+def test_shift_with_mirror_by_full_axis_length_reverses_axis(axis, polarity) -> None:
+    result = _apply_scalar_matrix(operators.shift_with_mirror(axis, SHAPE, polarity * SHAPE[axis]))
+    assert_close(result, numpy.flip(SCALAR_FIELD, axis=axis))
+
+
+@pytest.mark.parametrize('axis', [0, 1, 2])
+@pytest.mark.parametrize('shift_distance', [-1, 0, 1])
+def test_shift_with_mirror_on_singleton_axis_is_identity(axis, shift_distance) -> None:
+    shape = list(SHAPE)
+    shape[axis] = 1
+    field = numpy.arange(numpy.prod(shape)).reshape(shape)
+    result = (operators.shift_with_mirror(axis, shape, shift_distance) @ field.ravel()).reshape(shape)
+    assert_close(result, field)
+
+
 def test_vec_cross_matches_pointwise_cross_product() -> None:
     matrix_result = unvec(operators.vec_cross(vec(VECTOR_LEFT)) @ vec(VECTOR_RIGHT), SHAPE)
     expected = numpy.empty_like(VECTOR_LEFT)

meanas/test/test_fdmath_vectorization.py

@@ -1,4 +1,5 @@
 import numpy
+import pytest
 
 from ..fdmath import unvec, vec
 from ._test_builders import complex_ramp, real_ramp
@@ -44,3 +45,11 @@ def test_unvec_with_two_components_round_trips_vector() -> None:
 def test_vec_accepts_arraylike_input() -> None:
     arraylike = [[[1, 2], [3, 4]], [[5, 6], [7, 8]]]
     assert_close(vec(arraylike), numpy.ravel(arraylike, order='C'))
+
+
+@pytest.mark.parametrize('arraylike_type', [list, tuple])
+def test_unvec_accepts_arraylike_input(arraylike_type) -> None:
+    vector = numpy.arange(3 * numpy.prod(SHAPE)) * (1 + 0.5j)
+    field = unvec(arraylike_type(vector), SHAPE)
+    assert field.shape == (3, *SHAPE)
+    assert_close(vec(field), vector)

meanas/test/test_numerical_feedback.py

@@ -289,7 +289,10 @@ def test_slice_derivatives_match_periodic_reference_and_output_buffers(dtype, ax
         f = np.ascontiguousarray(f)
     widths = [np.linspace(.7, 1.3, n) for n in f.shape] if spaced else None
     derivative = (deriv_forward if forward else deriv_back)(widths)[axis]
-    expected = np.roll(f, -1, axis=axis) - f if forward else f - np.roll(f, 1, axis=axis)
+    # Widths promote the calculation before subtraction, including integer fields.
+    calculation_field = f.astype(np.result_type(f, widths[axis])) if spaced else f
+    expected = (np.roll(calculation_field, -1, axis=axis) - calculation_field if forward
+                else calculation_field - np.roll(calculation_field, 1, axis=axis))
     if spaced:
         shape = [1, 1, 1]
         shape[axis] = f.shape[axis]