Fch Gradient Flow Bench Eval

---
name: fch-gradient-flow-bench-eval
description: 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. Use when the user wants to benchmark on FCH Gradient Flow Benchmarks, or asks about evaluating this task. Reports L2 relative error.
metadata:
  skill_kind: dataset_eval
  source_arxiv: 2006.04784
  bibtex_key: christlieb2020fchbench
  confidence: high
---

# 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

```python
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

```bibtex
@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