Introduction

This material is based on the graduate course PHYS6350 Computational Physics that I taught at the University of Houston in Spring 2023 and Spring 2025

The course presents the basic concepts of computational physics and numerical methods, their examples, and applications to physics problems. The course is designed for graduate/advanced undergraduate physics students. The focus is on the concepts behind the numerical methods used in computational physics over the implementation. For this reason, most of the code is in Python and uses Jupyter Notebooks for presentation. For the same reason, many standard routines (such as linear equation solvers) are reimplemented.

The full course materials are available in the GitHub repository, including the sample code and slide decks.

This book consists of the following chapters:

• 1. Plotting basics
• 2. Machine precision
• 3. Interpolation
• 4. Linear algebra
• 5. Non-linear equations
  • Solving a non-linear equation
  • Polynomials and their roots
  • Systems of non-linear equations
  • Function minimization
• 6. Numerical integration
  • Basic methods
  • Adaptive methods
  • Improper integrals
  • High-order and Gaussian quadratures
• 7. Numerical differentiation
  • Finite differences
  • Automatic differentiation
  • Example: Susceptibility and Bose-Einstein condensation
• 8. Ordinary differential equations (ODEs)
  • Basic methods
  • Systems of ODEs
  • Time-reversal methods
  • Boundary value problems
• 9. Classical mechanics
  • Classical mechanics problems
  • Molecular dynamics simulations
• 10. Partial differential equations (PDEs)
  • Boundary value problems
  • Initial value problem: Heat equation
  • Initial value problem: Wave equation
• 11. Random numbers and Monte Carlo methods
  • Uniform random numbers
  • Computing integrals
  • Nonuniform random numbers
• 12. Statistical Physics
  • Markov chains and Metropolis algorithm
  • Ising model
  • Simulated annealing
  • Percolation threshold
• 13. Quantum Mechanics
  • Matrix methods for quantum mechanics
  • Time dependent Schrödinger equation
  • Variational Monte Carlo
• 14. Fourier transform