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

Python software and reproducible numerical examples for the paper Error attribution in gradient random walk methods for parabolic equations by Stephen Abkin and Prabir Daripa.

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

Clone the revised-paper branch:

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

For the archived version 1.1.0, download the ZIP from Zenodo, extract it, and open a terminal in Gradient-Random-Walk-Solvers-1.1.0/. If using the journal code supplement ZIP, open its source/ folder instead.

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:

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.

Reproduce the Paper

Generate all eleven figures directly from the committed data:

python reproduce.py paper

Rerun the representative simulations, refinement studies, ensembles, and two-step heat study, and compare their outputs with the committed results:

python reproduce.py verify-all

Reproduce the additional heat identity controls, independent seed-block controls, and fitted-rate bootstrap comparison:

python checks/heat_cdf_identity_check.py --fresh
python checks/two_step_reference_and_seed_blocks.py
python checks/bootstrap_seed_grouping_check.py --heat-profiles --json output/bootstrap.json

The bootstrap output's joint intervals are those reported in the manuscript. Figures are saved under output/final_prepublication_tests/paper_figures/. Rerunning replaces generated outputs. Allow several minutes for the full studies and several GB of available memory for the direct-distribution control. Runtime depends on hardware.

To run one study, use t4 for heat, t7 for paired heat reconstructions, t5 for the scalar reaction–diffusion front, t3 for Cole–Hopf plateau controls, t8 for Burgers controls, or t9 for two-step heat predictions and validation. For example:

python reproduce.py t9

Run python reproduce.py with no target to display the available commands.

Run a Modified Case

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

cp configs/heat_step_dirichlet.json configs/my_heat.json
python main.py configs/my_heat.json

Reaction–diffusion and Burgers examples are fhn_grw_steady.json and burgers_stationary_shock.json in configs/. config_template.jsonc documents the available fields. Plots are saved below outputs/. Save your edited input alongside them.

For a case matching one of the supplied exact references, also compute and print error metrics with:

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

The files in studies/ are complete examples of parameter sweeps, multi-seed experiments, error decompositions, and controlled comparisons. Adapt their parameters, seed lists, and output directories for new studies.

Repository Layout

• simulation.py, config.py, utils.py: solvers, configuration, and shared helpers.
• main.py, verify_solver.py: run a case and compare with an exact reference.
• configs/, config_template.jsonc: editable example inputs.
• studies/, study_paper_refinement.py: paper experiments and reusable study examples.
• reproduce.py, verify_ensembles.py, checks/: reproduce and check reported results.
• figure_data/, pinned_ensembles/, expected_values.json: reference data for the reported values and figures.
• figure_scripts/: figure generation.

Citation

Version 1.1.0, including the two-step heat study and updated checks, is archived on Zenodo. Cite this version when reproducing the revised paper. See CITATION.cff for author and paper 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.