Supplementary Material

These animations complement the static snapshots presented in Fig. 1. 2D Burgers benchmark and Fig. 2. LeVeque deformational flow benchmark of the manuscript.

Burgers' Equation

LeVeque Solution

This repository contains the Python implementation accompanying the paper:

A Control-Theoretic Framework on Moving-Mesh Methods for Solving PDEs

The repository includes two demonstration notebooks that reproduce the numerical experiments presented in the paper, together with animated visualizations of the corresponding PDE evolutions.

Repository Contents

.
├── 2DBurger_DEMO.ipynb      # Burgers' equation case study
├── LeVeque_DEMO.ipynb       # LeVeque case study
├── burgers.gif              # Animation of the Burgers' equation evolution
├── LeVeque.gif              # Animation of the LeVeque solution evolution
├──fig_burgers_grids.png     # Time Slice plot of the Burgers' equation evolution
├──fig_LeVeque_grids.png     # Time Slice plot of the LeVeque' equation evolution
└── README.md

Requirements

The notebooks require Python 3 together with the following packages:

• NumPy
• Matplotlib
• JAX
• Equinox
• Optax
• Diffrax

The following Python modules are used:

from __future__ import annotations
import os
import math
import pickle

import numpy as np
import matplotlib.pyplot as plt

import jax
import jax.numpy as jnp
import jax.scipy.linalg as jspla
from jax import lax, vmap, jit
import jax.tree_util as jtu

import equinox as eqx
import optax
import diffrax

Running the Examples

Open either notebook with Jupyter Notebook or JupyterLab and execute all cells.

• 2DBurger_DEMO.ipynb reproduces the results for the two-dimensional Burgers' equation.
• LeVeque_DEMO.ipynb reproduces the results for the LeVeque test case.

Animated Visualizations

The repository also contains animations of the complete temporal evolution shown in the paper:

• burgers.gif — evolution of the Burgers' equation solution along with the adaptive mesh.
• LeVeque.gif — evolution of the LeVeque solution along with the adaptive mesh.

License

This repository is released under the MIT License.