Trapezoidal conservation metric for 1D diffusion (restores strict tol, supersedes #566 loosening)

[ChrisRackauckas-Claude]

Summary

This supersedes the conservation-check tolerance loosening introduced in #566 with a proper grid-weighted (trapezoidal) quadrature, so the assertion in Test 03 of test/pde_systems/MOL_1D_Linear_Diffusion.jl holds at a strict tolerance again — honestly, without any fudge.

#566 (the OrdinaryDiffEq 7 / SciMLBase 3 / RecursiveArrayTools 4 migration) fixed length(sol) for single-variable PDE solutions, which made the per-timestep mass-conservation check actually run at every saved time instead of only at i = 1. Once it ran for the whole trajectory, the crude sum(u) rectangle-rule proxy was exposed: on the edge-aligned grid (nodes sit half a step outside the domain) the rectangle rule has an O(Δx) boundary bias and drifts to ~1e-7, so #566 relaxed atol from 1e-10 to 1e-6 with an explanatory comment. That was a band-aid; this PR removes it.

What changed

• Added a small trapezoidal_weights(xs) helper that builds composite trapezoidal weights from the actual grid spacing (each interior node gets the half-spacing on either side; the two endpoints get a single half-spacing). It is correct on the center-aligned grid, the half-step-offset edge-aligned grid, and non-uniform grids.
• Replaced @test sum(u_approx[i, :]) ≈ 0 atol = 1e-6 with @test dot(w, u_approx[i, :]) ≈ integral0 atol = 1e-9, where integral0 = dot(w, u_approx[1, :]) is the trapezoidal integral of the initial condition. Comparing to the conserved initial integral (rather than hardcoding 0) keeps the metric meaningful for any IC.

Why the strict tolerance is honest, not a refit

The rectangle-rule drift was a quadrature artifact, not a conservation defect of the scheme. The MoL semi-discretization with homogeneous Neumann BCs conserves the trapezoidal integral exactly at the spatial level; the only residual is the time integrator's error. Measured max|∫u dx| over the trajectory:

grid	Tsit5 tol = default (1e-6/1e-3)	Tsit5 tol = 1e-12/1e-12
center-aligned (300 pts)	4.13e-13	6.79e-16
edge-aligned (301 pts)	5.76e-10	7.87e-16

Tightening the solver tolerance drives the residual to machine precision on both grids, confirming it is bounded by time integration. For comparison the old sum(u) rectangle rule hit 9.8e-11 (center) / 1.10e-7 (edge) at the default tolerances.

The test uses default Tsit5 tolerances, so the realistic worst case is the edge-aligned 5.76e-10. The new atol = 1e-9 sits just above that (≈1.7x headroom on a deterministic computation) and is 1000x tighter than #566's 1e-6; the center-aligned grid passes with ~4 orders of margin. Both grid configurations (discretization and discretization_edge) are exercised by the existing for disc in [...] loop.

Local test evidence

$ GROUP=Diffusion julia +1.11 -e 'using Pkg; Pkg.test()'
...
Test Summary:                                      | Pass  Total      Time
MOLFiniteDifference Interface: 1D Linear Diffusion |  289    289  10m49.1s
     Testing MethodOfLines tests passed

289/289 pass with the strict atol = 1e-9, both grid alignments, no skips/@test_broken/loosening anywhere.

Stacking

This is stacked on #566 and includes its commits (notably the length/size solution-interface fix that makes this conservation check run at every timestep). It should land with or after #566. The diff unique to this PR is a single test file.

Notes

• Refs Bump the all-julia-packages group across 2 directories with 2 updates #564, Fix for OrdinaryDiffEq 7 / SciMLBase 3 / RecursiveArrayTools 4 (complements #565) #566.
• Please ignore until reviewed by @ChrisRackauckas.