Geometric Quasilinearization (GQL) | Kailiang Wu

Kailiang Wu
Kailiang Wu

Professor of Mathematics

Southern University of Science and Technology (SUSTech)

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Mailing address
Department of Mathematics, School of Science
Southern University of Science and Technology
1088 Xueyuan Avenue, Nanshan District
Shenzhen, Guangdong 518055, China

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Geometric Quasilinearization (GQL)

Geometric quasilinearization framework for analysis and design of bound-preserving schemes
K. Wu and C.-W. Shu
SIAM Review, Research Spotlight, 2023.
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@article{pub2023geometricquasilinearizationframeworkforanalysisanddesignofbound,
  author = {Kailiang Wu and Chi-Wang Shu},
  title = {{Geometric quasilinearization framework for analysis and design of bound-preserving schemes}},
  journal = {SIAM Review},
  year = {2023},
  volume = {65},
  number = {4},
  pages = {1031--1073},
  doi = {10.1137/21M1458247},
  eprint = {2111.04722},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1137/21M1458247}
}
Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations
K. Wu and H. Tang
Math. Models Methods Appl. Sci. (M3AS), 2017.
Paper arXiv
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@article{pub2017admissiblestatesandphysicalconstraintspreservingschemesforrelativistic,
  author = {Kailiang Wu and Huazhong Tang},
  title = {{Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations}},
  journal = {Math. Models Methods Appl. Sci. (M3AS)},
  year = {2017},
  volume = {27},
  number = {10},
  pages = {1871--1928},
  doi = {10.1142/S0218202517500348},
  eprint = {1603.06660},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1142/S0218202517500348}
}
Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics
K. Wu
SIAM Journal on Numerical Analysis, 2018.
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@article{pub2018positivitypreservinganalysisofnumericalschemesforidealmagnetohydrodynamics,
  author = {Kailiang Wu},
  title = {{Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics}},
  journal = {SIAM Journal on Numerical Analysis},
  year = {2018},
  volume = {56},
  number = {4},
  pages = {2124--2147},
  doi = {10.1137/18M1168017},
  eprint = {1802.02278},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1137/18M1168017}
}
Provably positive central discontinuous Galerkin schemes via geometric quasilinearization for ideal MHD equations
K. Wu*, H. Jiang, and C.-W. Shu
SIAM Journal on Numerical Analysis, 2023.
arXiv
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@article{pub2023provablypositivecentraldiscontinuousgalerkinschemesviageometricquasilinearization,
  author = {Kailiang Wu and Haili Jiang and Chi-Wang Shu},
  title = {{Provably positive central discontinuous Galerkin schemes via geometric quasilinearization for ideal MHD equations}},
  journal = {SIAM Journal on Numerical Analysis},
  year = {2023},
  volume = {61},
  pages = {250--285},
  eprint = {2203.14853},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2203.14853}
}
GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics
S. Ding and K. Wu*
Journal of Computational Physics, 2024.
Paper arXiv
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@article{pub2024gqlbasedboundpreservingandlocallydivergencefreecentral,
  author = {Shengrong Ding and Kailiang Wu},
  title = {{GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics}},
  journal = {Journal of Computational Physics},
  year = {2024},
  volume = {514},
  pages = {113208},
  doi = {10.1016/j.jcp.2024.113208},
  eprint = {2402.15437},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1016/j.jcp.2024.113208}
}
Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations
K. Wu* and C.-W. Shu
Numerische Mathematik, 2021.
arXiv
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@article{pub2021provablyphysicalconstraintpreservingdiscontinuousgalerkinmethodsformultidimensional,
  author = {Kailiang Wu and Chi-Wang Shu},
  title = {{Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations}},
  journal = {Numerische Mathematik},
  year = {2021},
  volume = {148},
  pages = {699--741},
  eprint = {2002.03371},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2002.03371}
}
Minimum principle on specific entropy and high-order accurate invariant region preserving numerical methods for relativistic hydrodynamics
K. Wu
SIAM Journal on Scientific Computing, 2021.
arXiv
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@article{pub2021minimumprincipleonspecificentropyandhighorderaccurate,
  author = {Kailiang Wu},
  title = {{Minimum principle on specific entropy and high-order accurate invariant region preserving numerical methods for relativistic hydrodynamics}},
  journal = {SIAM Journal on Scientific Computing},
  year = {2021},
  volume = {43},
  number = {6},
  pages = {B1164--B1197},
  eprint = {2102.03801},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2102.03801}
}
On local minimum entropy principle of high-order schemes for relativistic Euler equations
S. Cui, K. Wu*, and L. Xu
Mathematics of Computation, 2026.
Paper arXiv
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@article{pub2025onlocalminimumentropyprincipleofhighorderschemes,
  author = {Shumo Cui and Kailiang Wu and Linfeng Xu},
  title = {{On local minimum entropy principle of high-order schemes for relativistic Euler equations}},
  journal = {Mathematics of Computation},
  year = {2026},
  volume = {95},
  number = {362},
  pages = {2739-2787},
  doi = {10.1090/mcom/4139},
  eprint = {2503.12465},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1090/mcom/4139}
}
GQL-based physical-constraint-preserving high-order finite difference schemes for special relativistic hydrodynamics in arbitrary dimensions
L. Xu, S. Ding, and K. Wu*
Journal of Computational Physics, 2026. Accepted
arXiv
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@article{pub2026gqlbasedphysicalconstraintpreservinghighorderfinitedifferenceschemesforspecial,
  author = {Linfeng Xu and Shengrong Ding and Kailiang Wu},
  title = {{GQL-based physical-constraint-preserving high-order finite difference schemes for special relativistic hydrodynamics in arbitrary dimensions}},
  journal = {Journal of Computational Physics},
  year = {2026},
  eprint = {2606.31992},
  archivePrefix = {arXiv},
  note = {Accepted},
  url = {https://arxiv.org/abs/2606.31992}
}
Physical-constraint-preserving high-order finite volume WENO method via geometric quasilinearization for Chaplygin gas dynamics
L. Y. Win, Y. Chen*, Y. Nie, L. Cai, and K. Wu
Journal of Scientific Computing, 2025.
Paper
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@article{pub2025physicalconstraintpreservinghighorderfinitevolumewenomethodviageometricquasilinearization,
  author = {L. Y. Win and Y. Chen and Y. Nie and L. Cai and K. Wu},
  title = {{Physical-constraint-preserving high-order finite volume WENO method via geometric quasilinearization for Chaplygin gas dynamics}},
  journal = {Journal of Scientific Computing},
  year = {2025},
  volume = {105},
  pages = {33},
  doi = {10.1007/s10915-025-03059-w},
  url = {https://doi.org/10.1007/s10915-025-03059-w}
}
High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms
J. Wang, H. Tang, and K. Wu*
Journal of Computational Physics, 2024.
arXiv
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@article{pub2024highorderaccuratepositivitypreservingandwellbalanceddiscontinuous,
  author = {Jiangfu Wang and Huazhong Tang and Kailiang Wu},
  title = {{High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms}},
  journal = {Journal of Computational Physics},
  year = {2024},
  eprint = {2402.15446},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2402.15446}
}
A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations
S. Ding and K. Wu*
SIAM Journal on Scientific Computing, 2024.
Paper arXiv
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@article{pub2024anewdiscretelydivergencefreepositivitypreservinghighorder,
  author = {Shengrong Ding and Kailiang Wu},
  title = {{A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations}},
  journal = {SIAM Journal on Scientific Computing},
  year = {2024},
  volume = {46},
  number = {1},
  pages = {A50--A79},
  doi = {10.1137/23M1562081},
  eprint = {2305.14820},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1137/23M1562081}
}
A physical-constraint-preserving finite volume WENO method for special relativistic hydrodynamics on unstructured meshes
Y. Chen and K. Wu*
Journal of Computational Physics, 2022.
arXiv
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@article{pub2022aphysicalconstraintpreservingfinitevolumemethodforspecial,
  author = {Yaping Chen and Kailiang Wu},
  title = {{A physical-constraint-preserving finite volume WENO method for special relativistic hydrodynamics on unstructured meshes}},
  journal = {Journal of Computational Physics},
  year = {2022},
  volume = {466},
  pages = {111398},
  eprint = {2207.09385},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2207.09385}
}
Divergence-free finite volume WENO scheme for relativistic magnetohydrodynamics preserving positivity and subluminal velocity
S. Ding, K. Wu*, and C. Yuan
Monthly Notices of the Royal Astronomical Society (MNRAS), 2025.
Paper
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@article{pub2025divergencefreefinitevolumewenoschemeforrelativisticmagnetohydrodynamics,
  author = {S. Ding and K. Wu and C. Yuan},
  title = {{Divergence-free finite volume WENO scheme for relativistic magnetohydrodynamics preserving positivity and subluminal velocity}},
  journal = {Monthly Notices of the Royal Astronomical Society (MNRAS)},
  year = {2025},
  doi = {10.1093/mnras/staf280},
  url = {https://doi.org/10.1093/mnras/staf280}
}
Bound-preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme: One-dimensional case
R. Abgrall, M. Jiao, Y. Liu*, and K. Wu
Communications in Computational Physics, 2026.
Paper arXiv
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@article{pub2025boundpreservingpointaveragemomentpolynomialinterpretedpampaschemeonedimensionalcase,
  author = {R{\'e}mi Abgrall and Miaosen Jiao and Yongle Liu and Kailiang Wu},
  title = {{Bound-preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme: One-dimensional case}},
  journal = {Communications in Computational Physics},
  year = {2026},
  volume = {39},
  number = {1},
  pages = {29-58},
  doi = {10.4208/cicp.oa-2024-0266},
  eprint = {2410.14292},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.4208/cicp.oa-2024-0266}
}
Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes
S. Ding, S. Cui, R. Abgrall, and K. Wu*
arXiv, 2026. Preprint
arXiv
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@misc{pubarxiv260910476,
  author = {Shengrong Ding and Shumo Cui and Remi Abgrall and Kailiang Wu},
  title = {{Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes}},
  year = {2026},
  eprint = {2609.10476},
  archivePrefix = {arXiv},
  note = {Preprint},
  url = {https://arxiv.org/abs/2609.10476}
}
Provably positivity-preserving constrained transport scheme for 2D and 3D ideal magnetohydrodynamics
D. Pang and K. Wu*
Journal of Computational Physics, 2025.
arXiv
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@article{pub2025provablypositivitypreservingconstrainedtransportschemefor2dand3didealmagnetohydrodynamics,
  author = {Dongwen Pang and Kailiang Wu},
  title = {{Provably positivity-preserving constrained transport scheme for 2D and 3D ideal magnetohydrodynamics}},
  journal = {Journal of Computational Physics},
  year = {2025},
  volume = {541},
  pages = {114312},
  eprint = {2410.05173},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2410.05173}
}
Structure-preserving oscillation-eliminating discontinuous Galerkin schemes for ideal MHD equations: Locally divergence-free and positivity-preserving
M. Liu and K. Wu*
Journal of Computational Physics, 2025.
arXiv
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@article{pub2025structurepreservingoscillationeliminatingdiscontinuousgalerkinschemesforideal,
  author = {Mengqing Liu and Kailiang Wu},
  title = {{Structure-preserving oscillation-eliminating discontinuous Galerkin schemes for ideal MHD equations: Locally divergence-free and positivity-preserving}},
  journal = {Journal of Computational Physics},
  year = {2025},
  eprint = {2404.16794},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2404.16794}
}
Robust discontinuous Galerkin methods maintaining physical constraints for general relativistic hydrodynamics
H. Cao, M. Peng, and K. Wu*
Journal of Computational Physics, 2025.
arXiv
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@article{pub2025robustdiscontinuousgalerkinmethodsmaintainingphysicalconstraintsforgeneral,
  author = {Huihui Cao and Manting Peng and Kailiang Wu},
  title = {{Robust discontinuous Galerkin methods maintaining physical constraints for general relativistic hydrodynamics}},
  journal = {Journal of Computational Physics},
  year = {2025},
  eprint = {2410.05000},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2410.05000}
}
Bound-preserving OEDG schemes for Aw–Rascle–Zhang traffic models on networks
W. Chen, S. Cui*, K. Wu, and T. Xiong
Journal of Computational Physics, 2025.
Paper arXiv
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@article{pub2024boundpreservingoedgmethodforawrasclezhangtraffic,
  author = {Wei Chen and Shumo Cui and Kailiang Wu and Tao Xiong},
  title = {{Bound-preserving OEDG schemes for Aw--Rascle--Zhang traffic models on networks}},
  journal = {Journal of Computational Physics},
  year = {2025},
  volume = {520},
  pages = {113507},
  doi = {10.1016/j.jcp.2024.113507},
  eprint = {2409.16269},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1016/j.jcp.2024.113507}
}
Provably positivity-preserving, globally divergence-free central DG methods for ideal MHD system
R. Yan, H. Cao, and K. Wu*
Math. Models Methods Appl. Sci. (M3AS), 2026. Accepted
arXiv
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@article{pub2026provablypositivitypreservinggloballydivergencefreecentraldgmethodsforidealmhd,
  author = {Ruifang Yan and Huihui Cao and Kailiang Wu},
  title = {{Provably positivity-preserving, globally divergence-free central DG methods for ideal MHD system}},
  journal = {Math. Models Methods Appl. Sci. (M3AS)},
  year = {2026},
  eprint = {2508.08913},
  archivePrefix = {arXiv},
  note = {Accepted},
  url = {https://arxiv.org/abs/2508.08913}
}
An active-flux-type scheme for ideal MHD with provable positivity and a point-value-level discrete divergence-free constraint
M. Liu, D. Pang, R. Abgrall, and K. Wu*
Journal of Computational Physics, 2026. In press
Paper arXiv
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@article{pub2026anactivefluxtypeschemeforidealmhdwithprovablepositivityand,
  author = {Mengqing Liu and Dongwen Pang and Remi Abgrall and Kailiang Wu},
  title = {{An active-flux-type scheme for ideal MHD with provable positivity and a point-value-level discrete divergence-free constraint}},
  journal = {Journal of Computational Physics},
  year = {2026},
  doi = {10.1016/j.jcp.2026.115319},
  eprint = {2510.19721},
  archivePrefix = {arXiv},
  note = {In press},
  url = {https://doi.org/10.1016/j.jcp.2026.115319}
}