Add Burgers 1D FD playground for temporal order-of-convergence study with IMEX-SDC

[Copilot]

Tests temporal convergence order of IMEX-SDC on the 1D viscous Burgers equation using manufactured solutions (MMS), demonstrating full collocation order for homogeneous BCs, order reduction for inhomogeneous time-dependent BCs, and order restoration via boundary lifting.

New playground: pySDC/playgrounds/Burgers_1D_FD/

Problem classes (Burgers_1D_FD.py)

Three MMS classes sharing a base with a 4th-order FD Laplacian and 4th-order FD first-derivative operator (to keep spatial error floor ~1e-12):

• burgers_1d_hom — u_h = 0.1·sin(πx)·cos(t), homogeneous BCs. Autonomous f_impl = ν·A·u, no b_bc correction.
• burgers_1d_inhom — u_inh = 0.5 + 0.1·sin(πx)·cos(t) + 0.1x·sin(t), time-dependent right BC u(1,t) = 0.5 + 0.1·sin(t). f_impl = ν·A·u + ν·b_bc(t) with explicit time dependence.
• burgers_1d_inhom_lift — same exact solution as burgers_1d_inhom, reformulated with boundary lifting E(x,t) = 0.5 + 0.1x·sin(t). Lifted variable v = u − E satisfies homogeneous BCs with v_ex = 0.1·sin(πx)·cos(t). Autonomous f_impl = ν·A·v, no b_bc correction.

IMEX split: diffusion implicit (ν·A·u or ν·A·v), nonlinear advection + manufactured forcing explicit (-u·u_x + g, adjusted for lifting where applicable).

Convergence study (run_convergence.py)

M=3 RADAU-RIGHT, restol=1e-13, nvars=1023, ν=0.1, T_end=0.5:

Case	Observed orders	Expected
Homogeneous	4.96 → 5.03 → floor	5 (= 2M−1)
Inhomogeneous (b_bc)	3.55 → 3.74 → 3.92 → floor	~4 (= 2M−2)
Inhomogeneous (lift)	4.93 → 4.96 → floor	5 (= 2M−1)

The time-dependent right BC enters f_impl via b_bc(t), breaking the autonomous structure and reducing collocation order by 1. Boundary lifting removes b_bc from f_impl, restoring the full collocation order — consistent with the Allen-Cahn playground findings in #637.

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[pancetta]

@copilot Now add boundary lifting as #637 and check if this helps here, too.

[Copilot]

@copilot Now add boundary lifting as #637 and check if this helps here, too.

Added burgers_1d_inhom_lift in commit 6fab8ee. The lift is E(x,t) = 0.5 + 0.1x·sin(t), which makes v = u − E satisfy homogeneous BCs with v_ex = 0.1·sin(πx)·cos(t). The implicit operator is now the autonomous ν·A·v (no b_bc), and run_convergence.py now runs all three cases. Confirmed results:

• Homogeneous: order 4.96 → 5.03 (full order 5)
• Inhomogeneous (b_bc): order 3.55 → 3.74 → 3.92 (reduction to ~4)
• Inhomogeneous (lift): order 4.93 → 4.96 (full order 5 restored)