Numerical Methods for Time-Dependent Partial Differential Equations
Graduate Division
Computer Science

This course is intended as a topics course for graduate students in applied mathematics and computer science.  The course will focus on the behavior and numerical solution of hyperbolic partial differential equations (e.g., conservation laws).  If time permits, I will cover a few special topics towards the end of the semester.

Leveque R, "Finite Volume Methods for Hyperbolic Problems", Cambridge texts in applied mathematics, 2002.

Introduction to conservation laws -- advection, linear acoustics, elastic waves
Characteristics and the Riemann problem -- characteristic variables, domain of dependence, discontinuous solutions
Nonlinear conservation laws -- Burgers' equation, shocks, rarefaction waves, compression waves, vanishing viscosity limit, rankine-hugoniot condition, similarity solutions, entropy conditions
Finite volume methods -- conservative schemes, numerical flux, CFL condition, Godunov's method, Lax-Wendroff method, flux and slope limiters, TVD methods, transonic rarefactions, numerical viscosity, entropy conditions
Convergence, accuracy and stability -- local truncation errors, modified equations, lax-wendroff theorem
More nonlinear conservation laws -- shallow water equations, dam break problem, hugoniot loci, shock collision
More numerical methods -- Approximate Riemann solvers, Nonconvex fluxes, Fractional-step methods, Strang splitting
Multidimensional considerations -- finite difference methods, flux differencing methods, semidiscrete methods, CTU methods
Applications -- gas dynamics, elastic waves

calculus, linear algebra, introduction to numerical analysis

as seminar