Course Coordinator: 
Emile Touber
Nonlinear Waves: Theory and Simulations

Learn computational strategies to explore the rich and fascinating world of nonlinear wave phenomena. Through lectures and simulations, consider wave-related components and systems of balance laws (e.g. mass, momentum, energy) to identify what can trigger front-like structures such as the formation of shocks and solitons. Develop numerical models to explain wave-like motion by harnessing the explanatory power of hyperbolic partial differential equations (PDEs).


The target student has completed an engineering, physics or applied maths degree and is embarking on a PhD topic which involves in one way or another some nonlinear wave phenomena.
Course Content: 

Each week will be split into a theoretical and numerical component, as follows:

Theory (2 hours per week)

  1. Hyperbolic PDEs, characteristics
  2. Shockwaves: genesis, weak solutions, jump conditions
  3. Burgers’ equation
  4. Shock-boundary/-perturbation/-shock interactions
  5. Waves in networks
  6. Systems of balance laws
  7. Shocks in systems of hyperbolic PDEs
  8. Admissibility and stability of shocks
  9. Shock tubes
  10. Shock-refraction properties
  11. Extension to multiple dimensions
  12. Dispersive waves
  13. Dissipative solitons

Simulations (2 hours per week)

  1. Computer arithmetic, numerical chaos
  2. Time marching schemes, error types and their measurements
  3. Linear advection-diffusion equations, linear stability
  4. Burgers’ equation, non-linear stability, TVD and shock-capturing schemes
  5. Specifying and implementing well-posed boundary conditions
  6. Simulating traffic waves at a junction
  7. N-body simulations to measure macroscopic thermodynamic variables
  8. Solving the 1D Euler equation, notions of high-performance computing
  9. Solving the Riemann problem
  10. Solving shock-refraction problems
  11. Solving the 2D Euler equations, breakdown to turbulence
  12. Simulating a tidal bore
  13. Simulating biological patterns emerging from the Gray-Scott equations

Whilst the course is aimed at graduate students with an engineering/physics background, biologists interested in wave phenomena in biological systems (e.g. neurones, arteries, cells) are also welcome.

In-class notes are based on a number of excellent books, including but not limited to:

On waves

  • “Linear and nonlinear waves” by Whitham
  • “Nonlinear wave dynamics: complexity and simplicity” by Engelbrecht
  • “Waves in fluids” by Lighthill

On compressible flows

  • “Compressible-fluid dynamics” by Thompson
  • “Nonlinear waves in real fluids” by Kluwick

On continuum mechanics, systems of balance laws

  • “Non-equilibrium thermodynamics” by Groot
  • “Rational extended thermodynamics beyond the monatomic gas” by Ruggeri & Sugiyama
  • “Hyperbolic conservation laws in continuum physics” by Dafermos
  • “Systems of conservation laws” (2 volumes) by Serre

On solitary waves

  • “Solitons: an introduction” by Drazin
  • “Dissipative solitons in reaction diffusion systems” by Liehr

On computational methods

  • “A first course in computational fluid dynamics” by Aref & Balachandar
  • “Computational Gasdynamics” by Laney
  • “Shock-capturing methods for free-surface shallow flows” by Toro
  • “A shock-fitting primer” by Salas
Course Type: 
Individual project: first report: 25%, final report: 50%, final presentation: 25%
Text Book: 
  • In-class notes
Reference Book: 
    Prior Knowledge: 
     Prior knowledge of maths for engineers and physicists.