A Genuinely High Order Total Variation Diminishing Scheme for One-Dimensional Scalar Conservation Laws

Authors:
Xiangxiong Zhang;Chi-Wang Shu
Affiliations:
zhangxx@dam.brown.edu;shu@dam.brown.edu
Venue:
SIAM Journal on Numerical Analysis
Year:
2010

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Abstract

It is well known that finite difference or finite volume total variation diminishing (TVD) schemes solving one-dimensional scalar conservation laws degenerate to first order accuracy at smooth extrema [S. Osher and S. Chakravarthy, SIAM J. Numer. Anal., 21 (1984), pp. 955-984], thus TVD schemes are at most second order accurate in the $L^1$ norm for general smooth and nonmonotone solutions. However, Sanders [Math. Comp., 51 (1988), pp. 535-558] introduced a third order accurate finite volume scheme which is TVD, 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. By adopting the definition of the total variation for the numerical solutions as in [R. Sanders, Math. Comp., 51 (1988), pp. 535-558], it is possible to design genuinely high order accurate TVD schemes. In this paper, we construct a finite volume scheme which is TVD in this sense with high order accuracy (up to sixth order) in the $L^1$ norm. Numerical tests for a fifth order accurate TVD scheme will be reported, which include test cases from traffic flow models.