Error analysis of flux limiter schemes at extrema

Total variation diminishing (TVD) schemes have been an invaluable tool for the solution of hyperbolic conservation laws. One of the major shortcomings of commonly used TVD methods is the loss of accuracy near extrema. Although large amounts of anti-diffusion usually benefit the resolution of discontinuities, a balanced limiter such as Van Leer's performs better at extrema. Reliable criteria, however, for the performance of a limiter near extrema are not readily apparent. This work provides theoretical quantitative estimates for the local truncation errors of flux limiter schemes at extrema for a uniform grid. Moreover, the component of the error attributed to the flux limiter was obtained. This component is independent of the problem and grid spacing, and may be considered a property of the limiter that reflects the performance at extrema. Numerical test problems validate the results.