A modern approach to learning finite difference methods for partial differential equations, featuring Devito for high-performance PDE solvers.

Based on Finite Difference Computing with Partial Differential Equations by Hans Petter Langtangen and Svein Linge, this edition has been modernized with:

• Quarto for document generation (replacing DocOnce)
• Devito DSL for symbolic PDE specification and automatic code generation
• Modern Python practices with type hints, testing, and CI/CD

Devito is a domain-specific language (DSL) embedded in Python for solving PDEs using finite differences. Instead of manually implementing stencil operations, you write mathematical expressions symbolically and Devito generates optimized C code:

import numpy as np from devito import Constant, Eq, Grid, Operator, TimeFunction, solve # Define computational grid grid = Grid(shape=(101,), extent=(1.0,)) # Create field with time derivative capability u = TimeFunction(name='u', grid=grid, time_order=2, space_order=2) # Wave speed parameter (passed at runtime) c = Constant(name="c") # Set an initial condition (Gaussian pulse) x = np.linspace(0.0, 1.0, 101) u.data[0, :] = np.exp(-((x - 0.5) ** 2) / (2 * 0.1**2)) u.data[1, :] = u.data[0, :] # zero initial velocity (demo) # Write the wave equation symbolically and derive an explicit update stencil pde = Eq(u.dt2, c**2 * u.dx2) update = Eq(u.forward, solve(pde, u.forward)) # Devito generates optimized C code automatically op = Operator([update]) op.apply(time_M=100, dt=0.001, c=1.0)

# Clone the repository git clone https://github.com/devitocodes/devito_book.git cd devito_book # Create virtual environment and install dependencies python -m venv venv source venv/bin/activate # On Windows: venv\Scripts\activate pip install -e ".[devito]" # Build the book quarto render --to pdf

pip install -e ".[devito]" # With Devito support pip install -e ".[devito-petsc]" # With PETSc for implicit schemes pip install -e ".[dev]" # Development tools

Install Quarto:

# macOS brew install quarto # Ubuntu/Debian - see https://quarto.org/docs/get-started/ for current version # Download the .deb from https://github.com/quarto-dev/quarto-cli/releases/latest sudo dpkg -i quarto-*.deb

# macOS brew install --cask mactex # Ubuntu/Debian sudo apt-get install texlive-full

quarto render --to pdf # PDF only quarto render # All formats (HTML + PDF) quarto preview # Live preview with hot reload

Output: _book/Finite-Difference-Computing-with-PDEs.pdf

pytest tests/ -v # All tests pytest tests/ -v -m "not devito" # Skip Devito tests pytest tests/ -v -m devito # Devito tests only

1. Introduction to Devito - DSL concepts, Grid, Function, TimeFunction, Operator
2. Wave Equations - 1D/2D wave propagation, absorbing boundaries, sources
3. Diffusion Equations - Heat equation, stability analysis, 2D extension
4. Advection Equations - Upwind schemes, Lax-Wendroff, CFL condition
5. Nonlinear Problems - Operator splitting, Burgers' equation, Picard iteration

• Finite difference formulas and derivations
• Truncation error analysis
• Software engineering practices

devito_book/ ├── src/ # Python solvers and utilities │ ├── wave/ # Wave equation solvers (Devito) │ ├── diffu/ # Diffusion solvers (Devito) │ ├── advec/ # Advection solvers (Devito) │ ├── nonlin/ # Nonlinear solvers (Devito) │ ├── operators.py # FD operators with symbolic derivation │ └── verification.py # Symbolic verification utilities ├── tests/ # Pytest test suite ├── chapters/ # Quarto book chapters │ ├── devito_intro/ # Introduction to Devito │ ├── wave/ # Wave equations │ ├── diffu/ # Diffusion equations │ ├── advec/ # Advection equations │ ├── nonlin/ # Nonlinear problems │ └── appendices/ # Reference material ├── _quarto.yml # Book configuration └── pyproject.toml # Python package configuration 

1. Fork the repository
2. Create a feature branch: git checkout -b feature/improvement
3. Install dev dependencies: pip install -e ".[dev]"
4. Install pre-commit hooks: pre-commit install
5. Make changes and run tests: pytest tests/ -v
6. Verify the build: quarto render
7. Submit a Pull Request

Pre-commit hooks enforce:

• ruff - Linting
• isort - Import sorting
• typos - Spell checking
• markdownlint - Markdown formatting

This work is adapted from:

Langtangen, H.P., Linge, S. (2017). Finite Difference Computing with PDEs: A Modern Software Approach. Springer, Cham. DOI: 10.1007/978-3-319-55456-3

Licensed under CC BY 4.0.

Original Authors:

• Hans Petter Langtangen (1962-2016)
• Svein Linge

Adapted by:

• Gerard J. Gorman, Imperial College London

• Devito Project
• Devito API Reference
• Devito GitHub
• Original Book (PDF)