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

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.

Write $M_0={\sup }_x{u}_0(x)<\infty$ and assume $\lambda (1+p{M}_0^{p-1})\le 1$. Suprema are understood where the printed maxima on the infinite domain are not attained. We prove the invariant interval $0\le U_j^m\le M_0$ simultaneously with the stated speed bound.

Suppose the interval property holds at time $m$. By the mean value theorem, the divided differenceis between $1$ and $1+p{M}_0^{p-1}$; when the two states coincide define it as $f^{\prime }(U_j^m)$. The update is the convex combinationBoth coefficients are nonnegative and sum to one, so the same interval property holds at time $m+1$. It is true initially and hence for all times by induction. ThereforeThe propagated nonnegativity is essential; it keeps the flux increasing on the entire range of the numerical solution.

The convex-combination formula proved in part (i) gives $0\le U_j^{m+1}\le {\sup }_{\ell }{U}_{\ell }^m$ for every $j$. Taking the supremum and using nonnegativity givesThis is the discrete maximum principle for the nonlinear upwind step and proves the requested conditional ${\ell }^{\infty }$ stability under the propagated Courant–Friedrichs–Lewy condition.