Numerical Methods for Time-Dependent Partial Differential Equations

Rangan

Graduate Division

Computer Science

Description:

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.

Text:

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

Syllabus:

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

Prerequisites:

calculus, linear algebra, introduction to numerical analysis

Grading:

as seminar