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hyperbolic PDEs (Ch10) (

hyperbolic PDEs (Ch10)

2nd order time derivatives w/ 2 ICs

Hyperbolic conservation laws (Supp)

Theory (Ch1)

Expansion fans(Ch1.2)
pick the meaningful physical solution

Ex: Burgers'

Entropy solutions

Viscosity solutions \( \lim_{\epsilon \rightarrow 0}u^{\epsilon} \)
\( u^{\epsilon}_t+f(u^{\epsilon})_x -\epsilon u^{\epsilon}_{xx}=0\)

weak solutions
& Rankine-Hugoniot relation (Ch1.1)

weak solution in space-time

shock=
intersection of characteristic lines

speed = RH-relation \( s' \)
\( s'=\frac{f(u_l)-f(u_r)}{u_l-u_r} \)

position \( s(t) \)

Ex: Burgers'
\( s'=\frac{u_l+u_r}{2} \)

Intro (Ch0)

Examples

Burgers' eq
\( f(u) = \frac{1}{2} u^2 \)

Transport eq.

Euler eq.

Wave eq.

Setup
\( \frac{\partial u}{\partial t}+ \textrm{div}\, f(u)=0 \)

Numerical Methods (Ch2)

entropy-viscosity methods

Galerkin methods
(=natural ones)

Discontinuous

finite volume

space-time methods

up-wind like fluxes

space-discretization \( \Rightarrow \) 2nd order ODE

reduction to 1st order system of ODEs

introduce \( v=u' \)

Time integration methods for 1st order can be applied

examples: waves