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).

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

Theory (2 hours per week)

- Hyperbolic PDEs, characteristics
- Shockwaves: genesis, weak solutions, jump conditions
- Burgers’ equation
- Shock-boundary/-perturbation/-shock interactions
- Waves in networks
- Systems of balance laws
- Shocks in systems of hyperbolic PDEs
- Admissibility and stability of shocks
- Shock tubes
- Shock-refraction properties
- Extension to multiple dimensions
- Dispersive waves
- Dissipative solitons

Simulations (2 hours per week)

- Computer arithmetic, numerical chaos
- Time marching schemes, error types and their measurements
- Linear advection-diffusion equations, linear stability
- Burgers’ equation, non-linear stability, TVD and shock-capturing schemes
- Specifying and implementing well-posed boundary conditions
- Simulating traffic waves at a junction
- N-body simulations to measure macroscopic thermodynamic variables
- Solving the 1D Euler equation, notions of high-performance computing
- Solving the Riemann problem
- Solving shock-refraction problems
- Solving the 2D Euler equations, breakdown to turbulence
- Simulating a tidal bore
- 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

- In-class notes