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.

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%

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