Instructor:
Hans De Sterck
E-mail:
desterck@colorado.edu
Prerequisite: Permission of the instructor.
Boom. That was a sonic boom. A sonic boom?
A wide class of phenomena in daily life, as well as in more exotic physical systems of all scales, can be described by systems of partial differential equations that are of hyperbolic type. These phenomena have in common display of wavelike behavior and shocks. Small-amplitude waves behave in a linear fashion (e.g. sound waves), but large-amplitude waves can steepen into discontinuities or shocks due to nonlinear effects (yes, the sonic boom).
In this course, you will be introduced to the basic mathematical description of hyperbolic systems and their associated waves and shocks, and to basic techniques for the numerical solution of hyperbolic problems.
The first part of the course covers the mathematical theory of waves and shocks in hyperbolic systems, starting from the analysis of a simple scalar equation and going all the way to the system of Euler equations for gas dynamics. In the second part of the course, basic finite volume schemes will be introduced for the numerical simulation of hyperbolic problems with shocks in 1D and 2D. Implementation on parallel computers using message passing (MPI) is discussed. In the final part, more advanced applications are presented. The hyperbolic properties of the magnetohydrodynamic (MHD) description of a magnetized ionized gas (or plasma) are analyzed. The hyperbolic theory and simulation techniques are applied to problems of water waves, supersonic flight, aeronautics, solar mass ejections, planets, comets, and magnetic storms.
Course notes will be provided by the instructor. Students may choose among a variety of small assignments, with both analytical and computational problems.
I. Basic theory: Definition of hyperbolic systems and discussion of their wave properties. Characteristic curves. Linear waves, nonlinear waves, shocks, jump relations. Scalar Advection equation, Burgers equation, Shallow Water equations, Euler equations of gas dynamics.
II. Computation: Finite volume schemes for the numerical simulation of flows with shocks in 1D and 2D. Upwind schemes. Conservation, stability, dissipation, dispersion. Scalar and system schemes. Implementation on parallel computers using message passing (MPI).
III. Advanced applications: Magnetohydrodynamic (MHD) description of a magnetized plasma. MHD wave properties. MHD shocks. Numerical simulations of Shallow Water, Euler and MHD bow shock flows, with applications including water waves, supersonic flight, aeronautics, solar mass ejections, planets, comets, and magnetic storms.