3.2.4 Discontinuous initial data

• ``Standard" finite difference methods applied to linear systems work generally well for smooth solutions

• What happens when the initial data are discontinuous?

  Example:
  

  $$\frac{\partial u}{\partial t} + a\frac{\partial u}{\partial x} = 0 -\infty < x < \infty t\ge 0$$ (76)

  
  
  
  

  $$u_0(x) = \left\{ \begin{array}{ll} 1 & \mbox{if $x<0$} \\ 0 & \mbox{if $x>0$} \end{array} \right.$$ (77)

  
  

  The exact solution is $u_0(x-at)$. The numerical method can have difficulties near the discontinuity