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Numerical Solution of Conservation Laws
Last Updated: 2026-02-05 15:02:47
Abstract
Short course from 6 to 10 June 2005. For information and registration, please contact the secretary of the Institute of Fluid Dynamics or mail [email protected].
Objective
The objective of the short course is to introduce to the numerical solution of conservation laws. Prominent examples of conservation laws are the shallow water equations in hydraulics, the Euler and Navier-Stokes equations in fluid dynamics, the elasticity equations in solid mechanics, and the Maxwell equations in electrodynamics. If diffusion is neglected, conservation laws can be expressed as first order hyperbolic systems supporting waves and discontinuities like shocks. The short course is intended for graduate students and people from academia and industry, who want to learn the basics of the numerical solution of conservation laws.
Content
The basic mathematical properties of hyperbolic conservation laws will be outlined: characteristics, Rankine-Hugoniot condition, entropy condition, shocks, contact discontinuities and rarefaction waves. The Euler equations will serve to illustrate their physical meaning. The discretization of linear and nonlinear conservation laws by conservative finite difference methods and by finite volume methods will be explained. We shall consider the linear advection equation and the inviscid Burgers' equation as model equations. Examples of numerical flux functions will be given, e.g. for the upwind method and the Lax-Friedrichs method. We shall show how upwind discretization is related to central discretization plus numerical diffusion. Methods based on the exact or approximate solution of Riemann problems will be explained, namely Godunov's method and Roe's method. The development of high resolution methods will be outlined. Linear reconstruction and limiters will be used to obtain total variation diminishing (TVD) methods, which allow to capture shocks and contact discontinuities without numerical oscillations, but with higher accuracy than corresponding first order methods. Time integration by linear multistep methods and Runge-Kutta methods will be briefly described. The concepts of essentially non oscillatory (ENO) methods, weighted ENO (WENO) methods and discontinuous Galerkin (DG) methods will be mentioned to illustrate how essentially arbitrary high order of accuracy can be achieved for the numerical solution of conservation laws. The numerical methods will be generalized from scalar conservation laws to systems of conservation laws like the linearized and nonlinear 1D Euler equations. An outlook to the application to multidimensional conservation laws and the treatment of source terms will be mentioned.
General Information
- Language
- English
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- oral 30 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture |
Numerical Solution of Conservation Laws
Short course from 6 to 10 June 2005. Information and registration at the secretariate of the Institute of Fluid Dynamics or e-mail to
.
Blockkurs vom 6. bis 10. Juni 2005. Information und Anmeldung beim Sekretariat des Instituts für Fluiddynamik oder E-Mail an
.
|
|
15 h semesterly |