# Numerical Methods for Nonlinear Hyperbolic PDE

## News

Lecture December 9 **starts 10.00!**

All Problem sets are available.

New lecture scheduled: December 1, 10-15-12 in 2347

## Motivation

Nonlinear hyperbolic partial differential equations (PDEs) describe nonlinear wave motion and allow for discontinuities like shocks and contact discontinuities. Such equations arise in fluid dynamics, acoustics, elastodynamics, geophysics, astrophysics, and many other disciplines. In many applications, the equations are written as conservation laws where quantities such as mass and energy are conserved. Over the past years, appropriate numerical methods have been developed, applied and analyzed to cope with the challenges of nonlinear hyperbolic PDEs.

## Overview

The course treats the initial value problem for systems of hyperbolic equations such as the conservation law u_t + f(u)_x = 0 with the conserved quantity u = u(x,t). The main feature is the nonlinearity introduced by the flux function f(u). The solution can develop discontinuities in finite time, even when the initial data is smooth. Furthermore, it turns out that extra conditions have to be imposed to guarantee uniqueness. The course will describe finite volume approximations that can be used when standard approximation techniques break down, as they often do in the neighborhood of discontinuities.

## Contents

- scalar conservation laws
- entropy condition
- systems of hyperbolic PDEs
- shocks, rarefactions
- the Riemann problem
- first order finite volume approximations and Godunov's method
- second order TVD schemes
- higher order reconstructions like ENO and WENO
- systems of conservation laws like the Euler equations of gas dynamics
- numerical approximations for systems
- treatment of source terms
- multidimensional schemes

## Schedule

- Wednesday, September 30 2015, 10.15-12, 2244
- Wednesday, October 7 2015, 10.15-12, 1311
- Tuesday, October 13 2015, 10.15-12, 1245
- Wednesday, October 21 2015, 10.15-12, 2344
- Wednesday, October 28 2015, 10.15-12, 2345
- Wednesday, November 4 2015, 10.15-12, 2344
- Wednesday, November 11 2015, 10.15-12, 2345
- Wednesday, November 18 2015, 10.15-12, 2345
- Tuesday, December 1 2015, 10.15-12, 2347
- Wednesday, December 9 2015, 10.15-12, 2345

## Place

Polacksbacken, ITC, Uppsala University

## Examination

There will be theoretical and computational assignments after every second lecture. For 5 hp credit for the course, all assignments should be solved satisfactorily.

- Assignment 1 is due October 19
- Assignment 2 is due November 9
- Assignment 3 is due December 2
- Assignment 4 is due December 21

## References

*Numerical Methods for Conservation Laws* by Randall J. LeVeque, Birkhäuser Verlag, Basel, 1990.

*Finite Volume Methods for Hyperbolic Problems* by Randall J. LeVeque, Cambridge University Press, Cambridge, 2002.

## Lecturer

Gunilla Kreiss, Department of Information Technology, Uppsala University.

Please contact for more information. Per Lötstedt gave the course in 2011.