Add 1D Allen-Cahn FEniCS playground: SDC order reduction & boundary lifting

[Copilot]

Summary

New playground at pySDC/playgrounds/FEniCS/allen_cahn_1d/ demonstrating SDC order reduction with inhomogeneous time-dependent Dirichlet boundary conditions for the 1D Allen-Cahn equation, and its remedy via boundary lifting. This follows the blueprint of PR #632 (heat equation) but applied to Allen-Cahn.

Equation

$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \frac{2}{\varepsilon^2} u(1-u)(1-2u) - 6,d_w,u(1-u)$$

Exact solution (traveling wave):
$$u(x,t) = \tfrac{1}{2}!\left(1 + \tanh!\left(\frac{x - 0.5 - vt}{\sqrt{2},\varepsilon}\right)\right), \quad v = 3\sqrt{2},\varepsilon,d_w$$

The domain $[0,1]$ is a shift of $[-0.5, 0.5]$. Boundary conditions are inhomogeneous and time-dependent (exact solution evaluated at $x=0,1$).

New playground: pySDC/playgrounds/FEniCS/allen_cahn_1d/

Problem classes (problem_classes.py)

Both classes use an IMEX split with imex_1st_order_mass: diffusion implicit, nonlinear reaction explicit. Default parameters eps=0.3, dw=-0.04 ensure the front is well-resolved and the boundary values vary measurably with time.

• fenics_allencahn_imex_timebc — naive time-dependent BC imposition: bc.apply(b.values.vector()) inside solve_system overwrites boundary DOFs of the accumulated SDC right-hand side with the exact Dirichlet data at the new time, disrupting the sweeper's fixed-point and causing order reduction.

• fenics_allencahn_imex_timebc_lift — boundary lifting: decomposes $u = v + E$ where $E(x,t) = u_L(t)(1-x) + u_R(t),x$ is a linear interpolation of the boundary values. The transformed variable $v$ satisfies homogeneous BCs. solve_system only applies bc_hom (no overwrite), and eval_f includes the modified explicit forcing $M(-N(v+E) - E_t)$. This restores the full SDC collocation order.

Convergence study (run_convergence.py)

Standalone script varying dt over [0.5, 0.25, 0.125] and comparing estimated convergence orders for both problem classes.

Tests (test_order_reduction.py)

All tests are marked @pytest.mark.fenics.

Test	Assertion
test_allencahn_order_reduction	naive order < 2M − 1 − 0.3 for M=3
test_lifting_restores_full_order[M]	lifting order ≥ 2M − 2 for M ∈ {2, 3}
test_lifting_has_higher_order_than_naive	lifting order > naive order + 0.3

Security Summary

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[pancetta]

@copilot Did you introduce an artificial order reduction by applying the boundary conditions to the accumulated right-hand side in the solver? Is the order reduction gone if you don't do this?

[Copilot]

@copilot Did you introduce an artificial order reduction by applying the boundary conditions to the accumulated right-hand side in the solver? Is the order reduction gone if you don'...

No, the order reduction is not artificial. The previous implementation had a redundant second bc.apply(b.values.vector()) call, which I've now removed (commit d418a80).

The key point is that dolfin's bc.apply(T, b.values.vector()) call does two things at once: it modifies the system matrix and overwrites the boundary DOFs of b with the Dirichlet value u_D(t). This single necessary call is the source of the order reduction — it replaces whatever the SDC iteration had accumulated at the boundary nodes with the exact data at the new time, disrupting the sweeper's fixed-point property.

If you don't enforce the BCs at all in solve_system, the Dirichlet constraint is not applied and the solution is simply wrong. The only clean remedy — without compromising BC enforcement — is the boundary lifting in fenics_allencahn_imex_timebc_lift, which transforms to a homogeneous problem so that bc_hom.apply(T, b) sets boundary values to zero (consistent with what the SDC sweep has accumulated) and causes no disruption.