[maroba]

Improvements to TimeDependentPDE towards wave equations (#63), plus a fix for unstable time-dependent problems. Five commits that build on each other.

1. Faster implicit methods (ff09bd8)

The implicit steppers converted the full system matrix to LIL format in every step to inject boundary values, rebuilt the system whenever dt differed by round-off (as with np.linspace time points), and did not reuse the factorization. Now boundary rows are injected once with sparse masks, dt is compared with a relative tolerance, and the LU factorization is reused. Results are unchanged up to round-off; a Crank-Nicolson step on a 100x100 grid takes 0.8 ms instead of 148 ms.

2. Time-dependent boundary values and source terms (7b6a575)

bc[0] = lambda t: np.sin(2 * np.pi * t)        # also for Neumann and Robin
TimeDependentPDE(L, u0, bc, t, source=lambda t: ...)   # u_t = L(u) + s(t)

Values are evaluated at the times each method requires (including RK4 stages); with a manufactured solution all methods reach their order (Euler 1, RK4 4, Crank-Nicolson 2). PDE.solve() rejects time-dependent boundary conditions.

3. Second order in time (002f0f2)

Passing the initial velocity v0 solves u_tt = L(u) + s(t), e.g. the wave equation. New explicit method verlet (Störmer-Verlet, symplectic); rk4, crank-nicolson and backward-euler work via the equivalent first-order system (Crank-Nicolson is then equivalent to Newmark with β = 1/4, γ = 1/2). forward-euler is rejected (unstable for undamped waves). MOLSolution.v holds the velocities.

4. Stable second derivatives (60b3814)

While testing wave equations it turned out that one-sided high-order boundary stencils make time-dependent problems grow exponentially, regardless of the time stepper:

• Equidistant grids: complex eigenvalues (fatal for waves) from acc=4, eigenvalues with positive real part (fatal for diffusion) from acc=6 on. E.g. the heat equation with acc=6 blows up to 1e+68 instead of decaying.
• Non-uniform grids: since the wider central stencils of 0.14 (Fix accuracy-order loss for even derivatives on non-uniform grids #112), points next to the boundary used one-sided stencils starting at the point itself, which made even acc=2 unstable on Chebyshev grids.

Changes:

• New option Diff(..., boundary="centered") for equidistant grids: points near the boundary use the widest centered stencil that fits. The spectrum stays real and negative for any acc; the global error of time-dependent problems still converges with 4th–5th order (e.g. waves with acc=6: error 1e-5 instead of blow-up).
• Non-uniform grids: points near the boundary now use the stencil points closest to the boundary, including the boundary point. acc=2 and acc=4 are stable for diffusion and waves on Chebyshev, tanh and power-law grids, and mostly more accurate. Interior points and the boundary points themselves are unchanged.
• TimeDependentPDE checks the spectrum of each second derivative (1D, at most 201 points, a few ms) and warns if the problem would be unstable.

5. Docs (8645148)

• All Diff parameters documented in the API reference.
• Guides for time-dependent PDEs, boundary conditions and non-uniform grids updated; README and index; fixed a broken link to the compact schemes guide.
• The docs workflow now also builds on pull requests (without deploying).

Behavior changes (see CHANGES.md)

• Non-uniform grids: results at the points next to the boundary (not the boundary points themselves) change; they become more accurate.
• Equidistant grids: default unchanged; boundary="centered" is opt-in.

Test plan

• Full test suite passes locally (531 passed, 32 skipped); all new tests fail on the previous code
• Results of unchanged functionality compared numerically with the previous code (implicit steppers, static boundary conditions, interior of non-uniform operators)
• Docs built locally with Sphinx 9.2, no new warnings
• CI green
• Docs build in CI

[codecov]

Codecov Report

❌ Patch coverage is 98.10997% with 11 lines in your changes missing coverage. Please review.
✅ Project coverage is 95.18%. Comparing base (08c8455) to head (8645148).

Files with missing lines	Patch %	Lines
findiff/ivp.py	95.19%	7 Missing and 3 partials ⚠️
findiff/findiff.py	98.27%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #116      +/-   ##
==========================================
+ Coverage   94.87%   95.18%   +0.31%     
==========================================
  Files          42       43       +1     
  Lines        5915     6423     +508     
  Branches      342      399      +57     
==========================================
+ Hits         5612     6114     +502     
- Misses        251      254       +3     
- Partials       52       55       +3     

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