Novel Adaptive Schemes for Hyperbolic Conservation Laws
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910114213003264 |
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| author | Chu, Shaoshuai Feng, Pingyao Kolotilov, Vadim A. Kurganov, Alexander Ostapenko, Vladimir V. |
| author_facet | Chu, Shaoshuai Feng, Pingyao Kolotilov, Vadim A. Kurganov, Alexander Ostapenko, Vladimir V. |
| contents | 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 [{\sc S. Chu, A. Kurganov, and I. Menshov}, 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) finite-difference scheme, recently proposed in [{\sc V. A. Kolotilov, V. V. Ostapenko, and N. A. Khandeeva}, 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. Advantage of the resulting adaptive schemes are clearly demonstrated on a number of challenging numerical examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_18908 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Novel Adaptive Schemes for Hyperbolic Conservation Laws Chu, Shaoshuai Feng, Pingyao Kolotilov, Vadim A. Kurganov, Alexander Ostapenko, Vladimir V. Numerical Analysis 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 [{\sc S. Chu, A. Kurganov, and I. Menshov}, 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) finite-difference scheme, recently proposed in [{\sc V. A. Kolotilov, V. V. Ostapenko, and N. A. Khandeeva}, 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. Advantage of the resulting adaptive schemes are clearly demonstrated on a number of challenging numerical examples. |
| title | Novel Adaptive Schemes for Hyperbolic Conservation Laws |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2509.18908 |