Fch Gradient Flow Bench Eval

fch-gradient-flow-bench-eval

Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow — Christlieb et al. (2020) (arXiv:2006.04784, 2020)

What this evaluates

Evaluates the accuracy and stability of numerical schemes for solving the functionalized Cahn-Hilliard gradient flow under varying morphological complexity regimes. It tests how well different time-stepping methods (IMEX, SAV, PSD, ETD) capture defect formation, pearling, and interface dynamics in stiff, nonlinear PDE regimes.

Datasets

• FCH Gradient Flow Benchmarks — total ?; splits: test (5)

Metrics

• L2 relative error (primary) — range: other
  • Relative L2 norm error between the numerical scheme's solution and a reference solution at the final simulation time T. Computed as ||u_scheme - u_ref||_2 / ||u_ref||_2.

Input / output format

Input: Initial condition u(x,y,0) and PDE parameters (d, q, alpha_m, epsilon) defining the functionalized Cahn-Hilliard equation on a 2D domain.

Output: Numerical solution field u(x,y,T) at final time T, typically on a grid of size N x N (e.g., 256x256).

Scoring recipe

def compute_l2_relative_error(u_pred, u_ref):
    diff = u_pred - u_ref
    l2_norm = np.sqrt(np.sum(diff**2))
    ref_norm = np.sqrt(np.sum(u_ref**2))
    return l2_norm / ref_norm

Common pitfalls

• Requires high spatial resolution (N=512 or 1024) for stiff cases (Foot 2) to avoid grid-induced errors.
• Temporal tolerance (sigma_tol) must be tightened (e.g., 1e-6) for super-critical regimes to achieve quantitative agreement between schemes.
• Reference solution is typically the PSD scheme or a fixed small time-step solution, not an analytical ground truth.

Evidence (verbatim from paper)

We present an overview of the Benchmark simulations for local truncation error $\sigma_{\mathrm{tol}} = 10^{-5}$ , for which the PSD scheme is accurate while the IMEX, SAV, and ETD schemes are borderline accurate. Generically we find that a global $L^2 (\Omega)$ relative discretization error of $2.5\times 10^{-3}$ is sufficient to ensure that each scheme is quantitatively accurate, with the correct numbers, types, and placements of defects.

Citation

@misc{christlieb2020fchbench,
  title={Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow},
  author={Christlieb et al. (2020)},
  year={2020},
  note={arXiv:2006.04784}
}

• arXiv: 2006.04784