MA 347B: Hyperbolic Conservation Laws: Theory, Numerics, and Applications
Credits: 3:0
Prerequisites :
- Topics from MA222 are desirable but not absolutely necessary. A brief review of the essential concepts from measure theory will be provided in Chapter 3.
This course illustrates the interplay between mathematical theory and numerical analysis leading to an understanding of some of the famous real-world problems modeled by hyperbolic conservation laws. Main topics include:
- Linear Transport Equation: Method of characteristics; classical solutions; finite difference methods and their convergence analysis (Lax equivalence theorem).
- Scalar Conservation Laws: Method of characteristics; weak solutions; Rankine-Hugoniot condition; Riemann problems; mathematical entropy; initial value problems (IVPs); initial boundary value problems (IBVPs); Kruzkov’s uniqueness theorem for IVPs.
- L¹ space and BV Compactness: Helly’s Theorem.
- Numerical Methods and Existence Results for IVPs: a. Front Tracking Method: BV bounds and convergence analysis. b. Finite Volume Method: Monotone schemes; BV bounds and convergence analysis (Lax-Wendroff theorem). c. Splitting Method: Source splitting; dimension splitting; BV bounds and convergence analysis.
- Applications and Numerical Simulations: Traffic models (LWR models); two-phase flow in porous media (Buckley-Leverett model).
- System of Conservation Laws: Hyperbolicity, weak and entropy solutions; Lax-Liu entropy condition; Riemann problems; uniqueness results (statements only)
- Applications and Numerical Simulations: Dam-breaking problem (shallow water equations); chromatography equation; Euler equations.
Note: The course is primarily intended for third and fourth year B.Tech. Mathematics and Computing students, Integrated Ph.D. students in the Department of Mathematics, and BS Mathematics majors/minors. It may also serve as a foundational course on finite volume methods and hyperbolic conservation laws for M.Tech. and Ph.D. students interested in fluid dynamics. The course will be largely self-contained, with most of the required background covered or reviewed during the lectures.
Suggested books and references:
- A. Bressan, Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem, Oxford Lecture Ser. Math. Appl., Oxford University Press, Oxford, 2000.
- E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws [No. 3-4], Ellipses, Paris, 1991.
- H. Holden and N. H. Risebro, Front tracking for hyperbolic conservation laws, Springer, second ed., 2015.
- R. J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002.