Numerical methods for conservation laws, HW 1

1. Modify the upwinding, Lax-Friedrichs, Lax-Wendroff subroutines given in class (or obtained/written by yourself) to solve

\[\left\{ \begin{array}{ll} u_{t} + au_{x} = 0 \\ u(x, 0) = u_{0}(x)\\ u(0, t) = u(1, t) \end{array} \right.\]

on \((x, t) \in [0, 1] \times [0, 5]\)

  • Your code should take spatial stepsize ∆x, a, \(\lambda = a \Delta t\), and \(u_0(x)\) as input.

  • The output is a video showing the solution as time t goes from 0 to 5.

Solves Burgers equation using naive upwind scheme. For video run Matlab code attached.
Solves Burgers equation using Lax-Frierichs. For video run Matlab code attached.