Error estimates for Sanders third order accurate TVD scheme for scalar conservation laws in 1D

Abstract:   It is well known that a TVD finite volume scheme for 1D scalar conservation laws degenerate to first order accuracy at smooth extrema. But Sanders introduced a third order TVD scheme where the total variation is defined by measuring the variation of the reconstructed polynomials rather than the traditional way of measuring the variation of the grid values. He thus shows that for smooth and non-monotone solutions, his scheme would atmost degenerate to second order accuracy at extrema. In the first part of this talk, we will discuss the main idea behind Sanders scheme and discuss his results on accuracy and TVD properties of his scheme.       

Even though one can show accuracy rates for a finite volume Godunov type scheme, there has been no substantial literature available on L1 convergence of such high order accurate schemes. In the second part of this talk, we will prove second order L1 error estimates for the Sanders scheme.