Novel Adaptive Schemes for Hyperbolic Conservation Laws

Novel Adaptive Schemes for Hyperbolic Conservation Laws Full article

Journal

Communications in Computational Physics
ISSN: 1815-2406

Output data

Year: 2026, DOI: 10.4208/cicp.oa-2025-0256

Tags

Adaptive Schemes, Smoothness Indicator, Low-Dissipation Central-Upwind Numerical Fluxes, Overcompressive and Dissipative Limiters, Quasi-Linear Fifth-Order Finite-Difference Scheme, Euler Equations of Gas Dynamics

Authors

Chu Shaoshuai ¹ , Feng Pingyao ² , Kolotilov Vadim A. ³ , Kurganov Alexander ⁴ , Ostapenko Vladimir V. ³

Affiliations

1	Department of Mathematics, RWTH Aachen University, 52056 Aachen, Germany
2	Department of Mathematics, North Carolina State University, 27695 Raleigh, USA
3	Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
4	Department of Mathematics and Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, P.R. China

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.

Chu S. , Feng P. , Kolotilov V.A. , Kurganov A. , Ostapenko V.V.
Novel Adaptive Schemes for Hyperbolic Conservation Laws
Communications in Computational Physics. 2026. DOI: 10.4208/cicp.oa-2025-0256 OpenAlex

Published online:

Sep 29, 2026

≡ OpenAlex:

W4415251405