[fdmath] correct Gridlock Yee spacing order

Use center-to-center intervals for forward E derivatives and base-cell widths for backward H derivatives in the Gridlock examples. Document the distinction from meshes defined directly by forward intervals and fix the random fixture's backward averaging direction.

Add coordinate-derivative regression coverage for graded periodic and nonperiodic Gridlock meshes using both functional and sparse operators.

Validation: 1689 tests passed, 586 skipped; no new Ruff findings in changed Python files.

examples/eme.py

@@ -109,7 +109,7 @@ def build_geometry(
         z=Extent(min=0, max=1e6),
     )
 
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
     dxes_2d = [[d[0], d[1]] for d in dxes]
     return grid, epsilon, dxes_2d, omega
 

examples/eme_bend.py

@@ -99,7 +99,7 @@ def build_geometry(
         z=Extent(min=-1e6, center=0),
     )
 
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
     dxes_2d = [[d[0], d[1]] for d in dxes]
     return grid, epsilon, dxes_2d, omega
 

examples/fdfd0.py

@@ -67,7 +67,7 @@ def test0(solver=generic_solver):
         num_points = 24,
         )
 
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
     for a in (0, 1, 2):
         for p in (-1, 1):
             dxes = meanas.fdfd.scpml.stretch_with_scpml(dxes, axis=a, polarity=p, omega=omega,

examples/fdfd1.py

@@ -50,7 +50,7 @@ def test1(solver=generic_solver):
     epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
     grid.draw_cuboid(epsilon, x=dict(center=0, span=8e3), y=dict(center=0, span=w), z=dict(center=0, span=th), foreground=n_wg**2)
 
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
     for a in (0, 1, 2):
         for p in (-1, 1):
             dxes = scpml.stretch_with_scpml(dxes,omega=omega, axis=a, polarity=p,

examples/fdtd.py

@@ -141,7 +141,7 @@ def main() -> None:
     ee = numpy.zeros_like(epsilon, dtype=complex)
     hh = numpy.zeros_like(epsilon, dtype=complex)
 
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
 
     # PMLs in every direction
     pml_params = [

examples/tcyl.py

@@ -46,7 +46,7 @@ def test1(solver=generic_solver):
     epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
     grid.draw_cuboid(epsilon, center=center, dimensions=[8e3, w, th], foreground=n_wg**2)
 
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
     for a in (1, 2):
         for p in (-1, 1):
             dxes = scpml.stretch_with_scpml(

examples/waveguide.py

@@ -119,7 +119,7 @@ def main(
 
     # Add SCPML stretching to the FDFD grid; this matches the CPML-backed FDTD
     # run below so the two solvers see the same absorbing boundary profile.
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
     for a in (0, 1, 2):
         for p in (-1, 1):
             dxes = scpml.stretch_with_scpml(dxes, omega=omega, axis=a, polarity=p, thickness=pml_thickness)
@@ -269,7 +269,7 @@ def main2():
     ee = numpy.zeros_like(epsilon, dtype=complex)
     hh = numpy.zeros_like(epsilon, dtype=complex)
 
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes = [grid.autoshifted_dxyz(), grid.dxyz]
 
     # PMLs in every direction
     pml_params = [