An adaptive TVD limiter
When solving nonlinear hyperbolic conservation laws numerically one introduces limiters to enforce that the schemes are total variational diminishing, TVD, and of at least second order in regions of smooth solutions. In this work adaptive TVD limiters are studied which are based on a limiter approximating the upper boundary of the TVD range and that of the third-order upwind TVD scheme. The adaptive limiter switches to a compressive one near a discontinuity, to the third-order TVD scheme's limiter in smooth regions, and to a weighted averaged scheme in the transition region between smooth and high gradient solutions. The analysis is mainly done on a scalar one-dimensional conservation law. Following \textit{A. Harten}'s idea of a subcell resolution scheme [J. Comput. Phys. 83, No. 1, 148-184 (1989; Zbl 0696.65078)] the authors propose new rules for switching the limiters such that the discontinuities are better captured. Numerical examples show that the proposed scheme works very well for one-dimensional scalar equations but become less effective in one- and two-dimensional Euler equations of gas dynamics.
scientific article; zbMATH DE number 758629