Syllabus for a Graduate Course in

M9001: Computational Methods for Fluid Dynamics

Professor Thomas Y. Hou Charles Lee Powell Professor of Applied and Computational Mathematics

Applied Mathematics 217-50 California Institute of Technology

Pasadena, CA 91125 COURSE OUTLINE

1. Introduction 1.1. Review of basic approximation theory. 1.2. Numerical methods for ordinary differential equations. 1.3. Explicit methods versus implicit methods. Stability analysis.

2. Numerical methods for linear hyperbolic partial differential equations. 2.1. Finite difference approximations, von Neumann stability analysis. 2.2. High order schemes, multi-step methods 2.3. Convergence analysis 2.4. Boundary conditions.

3. Shock capturing methods for compressible flows. 3.1. Concept of weak solutions and entropy condition. 3.2. Godunov schemes, Riemann problems. 3.3. High resolution schemes.

4. Numerical methods for incompressible fluid flows. 4.1. The Projection method. 4.2. The pseudo-spectral methods. 4.3. Boundary conditions and error analysis.

5. Numerical methods for multi-phase flows. 5.1. Level set methods for multi-phase flows. 5.2. Immersed boundary methods. 5.3. The ghost fluid method.

LITERATURE

• John C. Strikwerda, “Finite Difference Schemes and Partial Differential Equations”, SIAM Publications, 2004.

• Randall J. LeVeque, “Finite Volume Methods for Hyperbolic Problems”, Cambridge University Press. 2002.

• J. A. Sethian, “Level Set Methods”, Cambridge University Press, 1996. • Papers by the lecturer

REQUIRED BACKGROUND Basic knowledge in numerical methods, partial differential equations, and fluid dynamics.

HOMEWORK ASSIGNMENTS Questions for book chapter, and assignments from class. FINAL PROJECT Literature Review project – submitted report and lecture in class.

SHORT BIO

Dr. Thomas Y. Hou is the Charles Lee Powell professor of applied and computational mathematics at Caltech, and is one of the leading experts in vortex dynamics and multiscale problems. His research interests are centered around developing analytical tools and effective numerical methods for vortex dynamics, interfacial flows, and multiscale problems. He was born in Guangzhou, China, and studied at the South China University of Technology before obtaining his Ph.D. from UCLA in 1987. Upon graduating from UCLA, he joined the Courant Institute as a postdoc and then became a faculty member in 1989. He moved to the applied math department at Caltech in 1993, and is currently the chair of the department of applied and computational mathematics. Dr. Hou has received a number of honors and awards for his academic achievements, which include the Computational and Applied Sciences Award from USACM in 2005, the Morningside Gold Medal in Applied Mathematics in 2004, the SIAM Wilkinson Prize in Numerical Analysis and Scientific Computing in 2001, the Frenkiel Award from the Division of Fluid Mechanics of APS in 1998, the Feng Kang Prize in Scientific Computing in 1997, and the Sloan Fellow from 1990 to 1992. He was also a plenary speaker at the International Congress of Industrial and Applied Mathematics in 2003 and an invited speaker of the International Congress of Mathematicians in 1998.