Gradient Random Walk Solvers for the Heat, FitzHugh–Nagumo, and Burgers' Equations

Python software and reproducible numerical examples for the paper On the Accuracy of Gradient Random Walk Methods for the Heat, FitzHugh–Nagumo, and Burgers' Equations by Stephen Abkin and Prabir Daripa (arXiv:2608.22592).

The repository supports two uses:

1. reproduce the reported tables and figures, and
2. modify the supplied configurations or study scripts to run new cases.

Install

For the archived version 1.0.0 release, download and extract the ZIP from Zenodo, then open a terminal in the extracted directory:

cd Gradient-Random-Walk-Solvers-1.0.0

Alternatively, clone the development repository:

git clone https://github.com/stephen122204/Gradient-Random-Walk-Solvers.git
cd Gradient-Random-Walk-Solvers
git checkout grw-solvers-v3

Create a Python 3.11 environment:

python -m venv .venv

Activate it on macOS or Linux:

source .venv/bin/activate

On Windows PowerShell, use:

.\.venv\Scripts\Activate.ps1

On Windows Command Prompt, use:

.venv\Scripts\activate.bat

Then install the pinned dependencies on any platform:

python -m pip install --upgrade pip
python -m pip install -r requirements.txt

The pinned environment uses Python 3.11.4. Generated files are written under output/ or outputs/. Both directories are ignored by Git.

Check the Installation

Confirm the environment works before the longer runs (takes a few seconds):

python -m unittest discover -s tests

Reproduce the Paper

Generate the ten figures used in the paper directly from the committed data:

python reproduce.py paper

Re-run the representative single-seed studies or all ensemble and controlled studies:

python reproduce.py studies
python reproduce.py ensembles

The individual ensemble targets are t4 (heat), t7 (paired heat-grid control), t5 (FitzHugh–Nagumo), t3 (Cole–Hopf plateau controls), and t8 (controlled Burgers attribution). For example:

python reproduce.py t8

The committed reference values can be checked with:

python reproduce.py verify
python reproduce.py verify-ensembles

Use python reproduce.py verify --deep to re-run the archived representative simulations as well as the tabulated studies. One command covers everything, the deep representative checks followed by a full re-run and comparison of the five ensemble studies:

python reproduce.py verify-all

Measured wall-clock times (Apple-Silicon laptop, pinned environment):

Command	What it covers	Wall clock
python reproduce.py paper	the paper's ten figures, from committed data	~6 s
python reproduce.py verify --deep	single-seed studies plus archived representative arrays (179 checks)	~17 s
python reproduce.py t3	Cole–Hopf plateau controls	~2 s
python reproduce.py t8	controlled Burgers attribution	~4 s
python reproduce.py t5	FitzHugh–Nagumo thirty-seed ensemble	~15 s
python reproduce.py t4	heat thirty-seed ensemble	~1.5 min
python reproduce.py t7	paired heat output-grid study	~1.5 min
python reproduce.py verify-all	release gate: deep checks plus all five ensemble studies, re-run and compared	~4 min

Allow longer on older hardware.

Run python reproduce.py with no target to display every available command.

Run a Modified Case

Copy a JSON file from configs/, change its parameters, and pass it to the solver:

python main.py configs/heat_step_dirichlet.json
python main.py configs/fhn_grw_steady.json
python main.py configs/burgers_stationary_shock.json

config_template.jsonc documents the available fields. Custom comparison figures are saved below outputs/. When an exact solution is available, a modified case can also be checked with:

python verify_solver.py --equation heat --config configs/heat_step_dirichlet.json

The files in studies/ are complete examples of parameter sweeps, multi-seed experiments, error decompositions, bootstrap intervals, and controlled comparisons. They can be copied and edited for new studies.

Repository Layout

• simulation.py, config.py: solver and configuration handling.
• configs/, config_template.jsonc: editable example inputs.
• studies/, study_paper_refinement.py: paper experiments and reusable study examples.
• reproduce.py: paper reproduction and verification entry point.
• figure_data/, pinned_ensembles/, expected_values.json: committed data behind the reported values and figures.
• figure_scripts/: figure generation.
• tests/: quick installation checks of core formulas and boundary operations.

Citation

Please cite the paper:

S. Abkin and P. Daripa, On the Accuracy of Gradient Random Walk Methods for the Heat, FitzHugh–Nagumo, and Burgers' Equations, arXiv preprint arXiv:2608.22592, 2026.

Version 1.0.0 of this software is archived on Zenodo: https://doi.org/10.5281/zenodo.22050659. See CITATION.cff for complete citation metadata.

Acknowledgments

The authors thank Oliver Stalker for providing an early version of the Python code.

Principal Investigator: Professor Prabir Daripa — Texas A&M University, Department of Mathematics

Other projects from the Daripa Research Group are available on the group's GitHub page.