Course Coordinator: 
Emile Touber
Nonlinear Waves: Theory and Simulations

Many physical processes exhibit some form of nonlinear wave phenomena. However diverse they are (e.g. from engineering to finance), however small they are (e.g. from atomic to cosmic scales), they all emerge from hyperbolic partial differential equations (PDEs). This course explores aspects of hyperbolic PDEs leading to the formation of shocks and solitary waves, with a strong emphasis on systems of balance laws (e.g. mass, momentum, energy) owing to their prevailing nature in Nature. In addition to presenting key theoretical concepts, the course is designed to offer computational strategies to explore the rich and fascinating world of nonlinear wave phenomena.

By the end of this course, participants dealing with wave-like phenomena in their research field of interest should be able to identify components that can trigger front-like structures (e.g. shocks, solitons) and be able to explore their motion numerically. 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. However, it is assumed that participants have prior knowledge of maths for engineers and physicists.


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. The potential audience is therefore broad in terms of its interest but is expected to know the basic maths and physics from the aforementioned degrees. Although the course includes a large theoretical component it is designed to build a core knowledge of computational approaches to solve nonlinear waves. It is therefore not suited to students only interested in mathematical approaches, and could for example appeal more to the bio-physicist or quantum physicist looking into building a numerical model of wave-like phenomena in biology or physics.
Course Content: 

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

Theory (2 hours per week)
01 Hyperbolic PDEs, characteristics
02 Shockwaves: genesis, weak solutions, jump conditions
03 Burgers’ equation
04 Shock-boundary/-perturbation/-shock interactions
05 Waves in networks
06 Systems of balance laws
07 Shocks in systems of hyperbolic PDEs
08 Admissibility and stability of shocks
09 Shock tubes
10 Shock-refraction properties
11 Extension to multiple dimensions
12 Dispersive waves
13 Dissipative solitons

Simulations (2 hours per week)
01 Computer arithmetic, numerical chaos
02 Time marching schemes, error types and their measurements
03 Linear advection-diffusion equations, linear stability
04 Burgers’ equation, non-linear stability, TVD and shock-capturing schemes
05 Specifying and implementing well-posed boundary conditions
06 Simulating traffic waves at a junction
07 N-body simulations to measure macroscopic thermodynamic variables
08 Solving the 1D Euler equation, notions of high-performance computing
09 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


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
Prior Knowledge: 

 Prior knowledge of maths for engineers and physicists.