Finite volume transport schemes

We analyze arbitrary order linear finite volume transport schemes and show asymptotic stability in L and L∞ for odd order schemes in dimension one. It gives sharp fractional order estimates of convergence for BV solutions. It shows odd order finite volume advection schemes are better than even order finite volume schemes. Therefore the Gibbs phenomena is controled for odd order finite volume schemes. Numerical experiments sustain the theoretical analysis. In particular the oscillations of the LaxWendroff scheme for small Courant numbers are correlated with its non stability in L. A scheme of order three is proved to be stable in L and L ∞.