Novel Adaptive Schemes for Hyperbolic Conservation Laws

Communications in Computational Physics

Abstract

We introduce new adaptive schemes for the one- and two-dimensional hyperbolic systems of conservation laws. Our schemes are based on an adaption strategy recently introduced in [S. Chu et al., Appl. Numer. Math., 209 (2025)]. As there, we use a smoothness indicator (SI) to automatically detect “rough” parts of the solution and employ in those areas the second-order finite-volume low-dissipation central-upwind scheme with an overcompressive limiter, which helps to sharply resolve nonlinear shock waves and linearly degenerate contact discontinuities. In smooth parts, we replace the limited second-order scheme with a quasi-linear fifth-order (in space and third-order in time) difference scheme, recently proposed in [V. A. Kolotilov et al., Comput. Math. Math. Phys., 65 (2025)]. However, direct application of this scheme may generate spurious oscillations near “rough” parts, while excessive use of the overcompressive limiter may cause staircase-like nonphysical structures in smooth areas. To address these issues, we employ the same SI to distinguish contact discontinuities, treated with the overcompressive limiter, from other “rough” regions, where we switch to the dissipative Minmod2 limiter. Advantages of the resulting adaptive schemes are clearly demonstrated on a number of challenging numerical examples.

Authors 5

1. Shaoshuai Chu Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University, 52056 Aachen, Germany

2. Pingyao Feng

   North Carolina State University

   Affiliation as printed

   Department of Mathematics, North Carolina State University, 27695 Raleigh, USA

3. В. А. Колотилов

   Russian Academy of Sciences · Siberian Branch of the Russian Academy of Sciences · Lavrentyev Institute of Hydrodynamics

   Affiliation as printed

   Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia

4. Alexander Kurganov

   Southern University of Science and Technology

   Affiliation as printed

   Department of Mathematics and Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, P.R. China

5. Владимир Викторович Остапенко

   Russian Academy of Sciences · Siberian Branch of the Russian Academy of Sciences · Lavrentyev Institute of Hydrodynamics

   Affiliation as printed

   Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia