Adaptive symmetric flux limiters with long computation times for hyperbolic conservation laws

The construction of the limiter is a critical factor in the traditional Total Variation Diminishing (TVD) scheme. Among the classical limiters, superbee has the lowest numerical dissipation, but it can lead to over-compression in smooth regions and excessive artificial steepening at discontinuities and critical points if the computation time is prolonged. Classical limiters like Minmod, van Leer, van Albada, and MC fail to distinguish between different wave types, and they can even cause numerical oscillations for multi-critical value problems with prolonged computation times. A class of adaptive limiters has been created by combining classical limiters with superbee. This adaptive limiter can achieve second-order accuracy in smoothed regions and effectively reduce over-compression and excessive artificial steepening for long computation times. Analytical and numerical results show that the MUSCL scheme with an adaptive limiter is efficient.