401-4766-16L  Topics in Mathematical and Computational Fluid Dynamics

SemesterSpring Semester 2016
LecturersS. Mishra, F. Weber
Periodicitynon-recurring course
Language of instructionEnglish

AbstractThe course will cover some essential advanced topics in fluid dynamics, from both a theoretical and numerical point of view. The proposed topics include theory for the incompressible Euler and Navier-Stokes equations and numerical methods to approximate them. Additional topics including theory and numerics for the compressible Euler equations may also be covered.
ObjectiveTo learn both theoretical aspects of PDEs governing fluid flows as well as numerical methods to approximate them.
Content1. Derivation of the PDEs governing fluid flows from first principles.
2. Theory for incompressible Navier-Stokes equation -- Leray-Hopf weak solutions, global existence. Regularity in two dimensions.
3. Theory for incompressible Euler equations: Well-posedness in two-space dimensions, vortex sheets, blow-up criteria in three dimensions. Non-uniqueness of admissible weak solutions.
4. Spectral and spectral viscosity methods for the Euler and Navier-Stokes equations and their convergence.
5. Finite difference projection methods.
6. Vortex methods for the incompressible Euler equations.
7. Measure valued and Statistical solutions.
If time permits, we also cover some topics on the Compressible Euler equations.
Lecture notesLast version of lecture notes of the course can be found here:
Prerequisites / NoticeA solid background in functional analysis, PDE and numerical methods for PDE.