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.

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

Prior knowledge of maths for engineers and physicists.