hyperbolic PDEs (Ch10)

2nd order time derivatives w/ 2 ICs

  • Hyperbolic conservation laws (Supp)

space-discretization ⇒ 2nd order ODE

examples: waves

Theory (Ch1)

reduction to 1st order system of ODEs

introduce \( v=u' \)

Intro (Ch0)

Numerical Methods (Ch2)

Time integration methods for 1st order can be applied

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

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

Examples

entropy-viscosity methods

Galerkin methods
(=natural ones)

space-time methods

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

up-wind like fluxes

Ex: Burgers'

weak solution in space-time

Entropy solutions

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

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

Transport eq.

Euler eq.

Discontinuous

shock=
intersection of characteristic lines

Wave eq.

finite volume

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

position \( s(t) \)

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