New Scheme Adaption Strategy for Hyperbolic Conservation Laws

Abstract

We introduce a new scheme adaption strategy for one- and two-dimensional hyperbolic systems of conservation laws. The proposed approach builds upon the adaptive framework introduced in [S. Chu, A. Kurganov, and I. Menshov, Appl. Numer. Math., 209 (2025), pp.155--170], where we first employed the smoothness indicator from [R. Lohner, Comput. Methods. Appl. Mech. Eng., 61 (1987), pp.323--338] to automatically detect ``rough'' and smooth parts of the computed solution, and then used different limiters in the detected regions. This adaptive strategy was based on a threshold needed to sharply separate ``rough'' and smooth regions. In this paper, we propose a different adaption strategy. We use SBM-type limiters and vary one of the limiting parameters continuously to allow a smooth transition between the ``rough'' and smooth areas. This way, compressive and overcompressive limiters are activated in the shock and contact wave vicinities only, while we gradually switch to dissipative limiters in the smooth regions. A series of one- and two-dimensional numerical tests for the Euler equations of gas dynamics demonstrates that the new scheme adaption strategy leads to a higher resolution and reduced numerical dissipation.