Past exam of the numerical solution of differential equations course of the University of Oxford / 2010 / A / 3 / a / Solution

On nonnegative states the characteristic speed is $f^{\prime }(u)=1+p{u}^{p-1}>0$, so upwind finite difference scheme uses the left spatial difference. With $\lambda =\Delta t/\Delta x$, the conservative upwind finite difference scheme isExplicitly the flux difference is $(U_j^m-U_{j-1}^m)+[(U_j^m{)}^p-(U_{j-1}^m{)}^p]$. Preservation of nonnegativity under the given Courant restriction, proved next, justifies the same left-upwind direction at every step.