Nonlinear Waves: Theory and Simulation

Course Aim

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 Description

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.

Course Contents

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

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


Individual project: first report: 25%, final report: 50%, final presentation: 25%

Prerequisites or Prior Knowledge

mathematics for engineers and physicists