Design and Analysis of Structure-preserving Methods: Bound/Positivity, Minimum Entropy Principle, Entropy Stability, Energy Stability, Well-balance, Asymptotic Preserving, Divergence-free

K. Wu, Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics, SIAM Journal on Numerical Analysis, 2018.

K. Wu and C.-W. Shu, Geometric quasilinearization framework for analysis and design of bound-preserving schemes, SIAM Review, 2023.

K. Wu* and C.-W. Shu, Provably positive high-order schemes for ideal magnetohydrodynamics: Analysis on general meshes, Numerische Mathematik, 2019.

K. Wu, Minimum principle on specific entropy and high-order accurate invariant region preserving numerical methods for relativistic hydrodynamics, SIAM Journal on Scientific Computing,  2021.

K. Wu*, H. Jiang, and C.-W. Shu, Provably positive central discontinuous Galerkin schemes via geometric quasilinearization for ideal MHD equations, SIAM Journal on Numerical Analysis, 2023.

S. Cui, S. Ding, and K. Wu*, On optimal cell average decomposition for high-order bound-preserving schemes of hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 2024.

Z. Sun, Y. Wei, and K. Wu*,  On energy laws and stability of Runge--Kutta methods for linear seminegative problems, SIAM Journal on Numerical Analysis, 2022.

K. Wu* and C.-W. Shu, Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations, Numerische Mathematik, 2021.

K. Wu and H. Tang, Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations, Math. Models Methods Appl. Sci. (M3AS), 2017.

K. Wu and Y. Xing, Uniformly high-order structure-preserving discontinuous Galerkin methods for Euler equations with gravitation: Positivity and well-balancedness, SIAM Journal on Scientific Computing,  2021.

K. Wu and C.-W. Shu, Entropy symmetrization and high-order accurate entropy stable numerical schemes for relativistic MHD equations, SIAM Journal on Scientific Computing, 2020.

K. Wu and C.-W. Shu, A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics, SIAM Journal on Scientific Computing, 2018.

K. Wu, Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics, Physical Review D, 2017.

K. Wu and H. Tang, High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics, Journal of Computational Physics, 2015.

H. Jiang, H. Tang, and K. Wu*, Positivity-preserving well-balanced central discontinuous Galekin schemes for the Euler equations under gravitational fields, Journal of Computational Physics, 2022.

S. Cui, S. Ding, and K. Wu*, Is the classic convex decomposition optimal for bound-preserving schemes in multiple dimensions?, Journal of Computational Physics, 2023.

A. Chertock, A. Kurganov, M. Redle, and K. Wu, A new locally divergence-free path-conservative central-upwind scheme for ideal and shallow water magnetohydrodynamics, preprint, 2022.
W. Chen, K. Wu, and T. Xiong, High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers, Journal of Computational Physics, 2023.

Y. Ren, K. Wu, J. Qiu, and Y. Xing, On high-order positivity-preserving well-balanced finite volume methods for the Euler equations with gravitation, Journal of Computational Physics, 2023. 

C. Cai, J. Qiu, and K.Wu*, Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics, Journal of Computational Physics, 2024.

L. Xu, S. Ding, and K. Wu*, High-order accurate entropy stable schemes for relativistic hydrodynamics with a general equation of state, Journal of Scientific Computing, 2024. 

S. Ding and K. Wu*, A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations, SIAM Journal on Scientific Computing, 2024.

W. Chen, K. Wu, and T. Xiong, High order structure-preserving finite difference WENO scheme for MHD equations with gravitation in all sonic Mach numbers, Journal of Scientific Computing, 2024. 
S. Cui, A. Kurganov, and K. Wu*, Bound-preserving framework for central-upwind schemes for general hyperbolic conservation laws, preprint, 2024.
C. Cai, J. Qiu, and K. Wu*, Efficient provably convergent Newton-Raphson method for primitive variables in relativistic MHD equations, preprint, 2024.

Deep Learning and Data-driven Modeling

J. Chen and K. Wu*, Positional knowledge is all you need: Position-induced Transformer (PiT) for operator learning, International Conference on Machine Learning (ICML), 2024.

J. Chen and K. Wu*, Deep-OSG: Deep learning of operators in semigroup, Journal of Computational Physics,  2023.

C. Zhang, K. Wu, and Z. He, Critical sampling for robust evolution operator learning of unknown dynamical systems, IEEE Transactions on Artificial Intelligence, 2024. 

K.Wu and D. Xiu, Data-driven deep learning of partial differential equations in modal space, Journal of Computational Physics, 2020.

T. Qin, K. Wu, and D. Xiu, Data driven governing equations approximation using deep neural networks, Journal of Computational Physics, 2019.

K. Wu, T. Qin, and D. Xiu, Structure-preserving method for reconstructing unknown Hamiltonian systems from trajectory data, SIAM Journal on Scientific Computing, 2020.

Z. Chen, V. Churchill, K. Wu, and D. Xiu, Deep neural network modeling of unknown partial differential equations in nodal space, Journal of Computational Physics, 2022.

Z. Chen, K. Wu, and D. Xiu, Methods to recover unknown processes in partial differential equations using data, Journal of Scientific Computing, 2020.

J. Hou, T. Qin, K. Wu and D. Xiu,  A non-intrusive correction algorithm for classification problems with corrupted data, Commun. Appl. Math. Comput., 2020.

K. Wu and D. Xiu, Numerical aspects for approximating governing equations using data, Journal of Computational Physics, 2019.

Mathematical Properties and High-order Numerical Methods of Relativistic Hydrodynamic Equations

K. Wu, Minimum principle on specific entropy and high-order accurate invariant region preserving numerical methods for relativistic hydrodynamics, SIAM Journal on Scientific Computing,  2021.

K. Wu, Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics, Physical Review D, 2017.

K. Wu and H. Tang, High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics, Journal of Computational Physics, 2015.

K. Wu and H. Tang, Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations, Math. Models Methods Appl. Sci. (M3AS), 2017.

K. Wu* and C.-W. Shu, Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations, Numerische Mathematik, 2021.

K. Wu and H. Tang, A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics, SIAM Journal on Scientific Computing, 2016.

K. Wu and H. Tang, Physical-constraint-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state, Astrophys. J. Suppl. Ser. (ApJS), 2017.

K. Wu and H. Tang, Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics, Journal of Computational Physics, 2014.

Y. Chen and K. Wu*, A physical-constraint-preserving finite volume method for special relativistic hydrodynamics on unstructured meshes, Journal of Computational Physics, 2022.

L. Xu, S. Ding, and K. Wu*, High-order accurate entropy stable schemes for relativistic hydrodynamics with a general equation of state, Journal of Scientific Computing, 2024. 

C. Cai, J. Qiu, and K.Wu*, Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics, Journal of Computational Physics, 2024.

C. Cai, J. Qiu, and K. Wu*, Efficient provably convergent Newton-Raphson method for primitive variables in relativistic MHD equations, preprint, 2024.

High-dimensional Function Approximation：Big Data, Optimal Sampling, and Polynomial Approximation

K. Wu, Y. Shin, and D. Xiu, A randomized tensor quadrature method for high dimensional polynomial approximation, SIAM Journal on Scientific Computing, 2017.

K. Wu and D. Xiu, Sequential function approximation on arbitrarily distributed point sets, Journal of Computational Physics, 2018.

Y. Shin, K. Wu, and D. Xiu, Sequential function approximation with noisy data, Journal of Computational Physics, 2018.

K. Wu and D. Xiu,  Sequential approximation of functions in Sobolev spaces using random samples, Commun. Appl. Math. Comput., 2019.

Generalized Riemann Problem Solvers and Godunov-type Schemes for Hyperbolic Conservation Laws

K. Wu and H. Tang, A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics, SIAM Journal on Scientific Computing, 2016.

K. Wu and H. Tang, Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics, Journal of Computational Physics, 2014.

K. Wu, Z. Yang, and H. Tang, A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics, Journal of Computational Physics, 2014.

Uncertainty Quantification and Stochastic Galerkin Methods

K. Wu, D. Xiu, and X. Zhong, A WENO-based stochastic Galerkin scheme for ideal MHD equations with random inputs, Communications in Computational Physics, 2021.

K. Wu, H. Tang, and D. Xiu, A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty, Journal of Computational Physics, 2017.

Teaching（授课课程）

Fall Semester：Mathematical Experiments 数学实验（本科生）

Spring Semester：Computational Fluid Dynamics and Deep Learning 计算流体力学与深度学习（本研）

2021 Spring Semester：Calculus II 高等数学II（本科生）

2022 Spring Semester：Advanced Topics in Modern Computational Mathematics （本科生）

Group Members (课题组成员)

Research Assistant Professor（研究助理教授）

• Dr. Shumo Cui (2023.2.1-):  Ph.D. from Tulane University;  Postdoc at Temple University; VAP at SUSTech. 2024年获得主持国家自然科学基金-面上项目. We have a joint article: "Is the classic convex decomposition optimal for bound-preserving schemes in multiple dimensions?" published in 《Journal of Computational Physics》. We have a joint article: "On optimal cell average decomposition for high-order bound-preserving schemes of hyperbolic conservation laws" accepted for publication in 《SIAM Journal on Numerical Analysis》. We have a joint article with Prof. Alexander Kurganov: "Bound-preserving framework for central-upwind schemes for general hyperbolic conservation laws" accepted by 《SIAM Journal on Scientific Computing》. We have a joint article with Prof. Tao Xiong: "Bound-preserving OEDG method for Aw-Rascle-Zhang traffic models on networks" submitted to SISC.
• Dr. Shengrong Ding (2023.12-): Ph.D. from University of Science and Technology of China（中科大博士）; Postdoc at SUSTech（南科大博士后）.  2024年获得主持国家自然科学基金-青年项目.  We have a joint article: "Is the classic convex decomposition optimal for bound-preserving schemes in multiple dimensions?" published in 《Journal of Computational Physics》. We have a joint article: "On optimal cell average decomposition for high-order bound-preserving schemes of hyperbolic conservation laws" accepted for publication in 《SIAM Journal on Numerical Analysis》. We have a joint article "A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations" published in 《SIAM Journal on Scientific Computing》. We have a joint article: "High-order accurate entropy stable schemes for relativistic hydrodynamics with a general equation of state" published in《Journal of Scientific Computing》. We have a joint article: "GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics" published in 《Journal of Computational Physics》(A special issue in honor of 俄罗斯科学院院士 Sergei Godunov). 

Postdoctoral Fellows（博士后研究员）

• Dr. Shengrong Ding (2021.11-2023.11): Ph.D. from University of Science and Technology of China（中科大博士）. We have a joint article: "Is the classic convex decomposition optimal for bound-preserving schemes in multiple dimensions?" published in 《Journal of Computational Physics》. We have a joint article: "On optimal cell average decomposition for high-order bound-preserving schemes of hyperbolic conservation laws" accepted for publication in 《SIAM Journal on Numerical Analysis》. We have a joint article "A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations" published in 《SIAM Journal on Scientific Computing》. We have a joint article: "High-order accurate entropy stable schemes for relativistic hydrodynamics with a general equation of state" published in《Journal of Scientific Computing》. We have a joint article: "GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics" published in 《Journal of Computational Physics》(A special issue in honor of 俄罗斯科学院院士 Sergei Godunov). 

• Dr. Junfeng Chen (Postdoc Fellow: 2022.10-present; Visiting Postdoc Scholar: 2022.03-2022.09): B.Sc. from Tsinghua University（清华大学本科），Ph.D. from Paris Sciences et Lettres – PSL Research University（法国巴黎文理研究大学博士）. We have a joint article "Deep-OSG: Deep learning of operators in semigroup" published in《Journal of Computational Physics》. We have a joint article "Positional knowledge is all you need: Position-induced Transformer (PiT) for operator learning" published in 《International Conference on Machine Learning (ICML)》.  We have a joint article "DUE: A deep learning framework and library for modeling unknown equations" submitted. 

• Dr. Ruifang Yan (Postdoc Fellow: 2023.07-): Ph.D. from Wuhan University（武汉大学博士）. We have a joint article "Uniformly high-order bound-preserving OEDG schemes for two-phase flows" published in M3AS. 

• Dr. Huihui Cao (Postdoc Fellow: 2023.07-): Ph.D. from Xiangtan University（湘潭大学博士）. We are writing a joint article "Robust OEDG schemes for general relativistic hydrodynamics". 

• Dr. Chuan Fan (Postdoc Fellow: 2023.09-; Visiting Postdoc Scholar: 2023.06-2023.09): Ph.D. from Xiamen University（厦门大学博士）. We have a joint article "High-order oscillation-eliminating Hermite WENO method for hyperbolic conservation laws" published in 《Journal of Computational Physics》. 

• Dr. Mengqing Liu (Postdoc Fellow: 2023.09-): Ph.D. from University of Chinese Academy of Sciences（中国科学院大学博士）. We have a joint article "Structure-preserving oscillation-eliminating discontinuous Galerkin schemes for ideal MHD equations: Locally divergence-free and positivity-preserving" submitted to JCP.

• Dr. Qinghe Wang (Postdoc Fellow: 2023.12-): Ph.D. from the Chinese University of Hong Kong, Shenzhen（香港中文大学深圳分校博士）. We are working on deep learning for differential equations and operator learning. 

• Dr. Caiyou Yuan (Postdoc Fellow: 2024.09-)：Ph.D. from Peking University（北京大学博士）.
  

Graduate Students (研究生)

• Haili Jiang (2021.04-2021.12)，Visiting Ph.D. student from Peking University（北京大学）. We have a joint article with Prof. Chi-Wang Shu: "Provably positive central DG schemes via geometric quasilinearization for ideal MHD equations" published in《SIAM Journal on Numerical Analysis》. We have a joint article with Prof. Huazhong Tang: "Positivity-preserving well-balanced central discontinuous Galerkin schemes for the Euler equations under gravitational fields" published in《Journal of Computational Physics》.

• Fang Yan (2021.09-2023.07)，Master student，B.Sc. from South China University of Technology（华南理工）.

• Zhuoyun Li (2022.09-)，Ph.D. student，B.Sc. from SUSTech（南科大）. We have a joint article "Spectral volume from a DG perspective: oscillation elimination, stability, and optimal error estimates" submitted. 

• Manting Peng (2022.09-)，Master student，B.Sc. from SUSTech（南科大）. We have a joint article: "OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws" published in 《Mathematics of Computation》（计算数学顶级期刊）. We are writing several papers, including one on OECDG, one on "Robust OEDG schemes for general relativistic hydrodynamics", and one on "optimal convergence of ALE DG". 

• Linfeng Xu (2022.09-)，Master student，B.Sc. from SUSTech（南科大）. We have a joint article: "High-order accurate entropy stable schemes for relativistic hydrodynamics with a general equation of state" published in《Journal of Scientific Computing》. We are writing a joint article "Provably entropy-bounded high-order schemes for relativistic hydrodynamics with Synge-type equation of state". 

• Dongwen Pang (2023.09-), Ph.D. student, B.Sc. from Wuhan University of Technology（武汉理工）, M.Sc. from Xiangtan University（湘潭大学）. He is a visiting master student from 2023.03 to 2023.08.  We are writing a joint article "A second-order provably positivity-preserving constrained-transport with globally discrete zero-divergence for ideal MHD".

• Miaosen Jiao (2023.09-), Master student, B.Sc. from SUSTech（南科大）. We are working on structure-preserving active flux method for hyperbolic conservation laws.

• Caiyou Yuan(2023.06.29-08.30), Visiting Ph.D. student from Peking University（北京大学）. We are writing a joint article with Prof. Huazhong Tang on "Bound-preserving adaptive moving mesh method for special relativistic hydrodynamics".

• Zhihao Zhang (2023.06.29-08.30), Visiting Ph.D. student from Peking University（北京大学）. We have a joint article with Prof. Huazhong Tang: "High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: Well balancedness and positivity" submitted to JCP.

• Jiangfu Wang (2023.06.29-08.30), Visiting Ph.D. student from Peking University（北京大学）. We have a joint article with Prof. Huazhong Tang: "High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms" submitted to 《Journal of Computational Physics》. 

• Wei Chen, Visiting Ph.D. student from Xiamen University. We have a joint article with Prof. Tao Xiong: "High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers" published in 《Journal of Computational Physics》.  We have a joint article with Prof. Tao Xiong: "High order structure-preserving finite difference WENO scheme for MHD equations with gravitation in all sonic Mach numbers" published in 《Journal of Scientific Computing》. We have a joint article with RAP Shumo Cui and Prof. Tao Xiong: "Bound-preserving OEDG method for Aw-Rascle-Zhang traffic models on networks" submitted to 《SIAM Journal on Scientific Computing》.

• Chaoyi Cai, Visiting Ph.D. student from Xiamen University. We have a joint article with Prof. Jianxian Qiu: "Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics" published in 《Journal of Computational Physics》. We have a joint article with Prof. Jianxian Qiu: "Provably convergent and robust Newton-Raphson method: A new dawn in primitive variable recovery for relativistic MHD" submitted to 《SIAM Journal on Numerical Analysis》.
  

Undergraduate Students (本科生)

• Yuanzhe Wei: We have a joint article with Prof. Zheng Sun: "On energy laws and stability of Runge-Kutta methods for linear seminegative problems" published in 《SIAM Journal on Numerical Analysis》（计算数学方向的顶级期刊，南科大本科生首次）。He was selected to an exchange study program in MIT（麻省理工）南科大数学系第一位入选MIT交流项目的学生，见报道 [https://mp.weixin.qq.com/s/nhlTvmGpdOrXuwZ-a7v4Tg] . He won SUSTech outstanding bachelor thesis (本科毕业论文入选南科大优秀毕业论文). He received two offers from MIT and one offer from Brown University（收获麻省理工的2个offer和布朗大学的1个offer）; He chooses to pursue his Ph.D. at Brown University（布朗大学）since Sep. 2023.

• Yunhao Jiang: He was selected into a joint study program in University of Wisconsin-Madison（威斯康星大学麦迪逊分校）. He won 3rd class prize in the 2022 International Mathematics Competition for University Students (国际大学生数学竞赛).

• Zhuoyun Li: 推免研究生（GPA排名并列第一）. He won SUSTech outstanding bachelor thesis (本科毕业论文入选南科大优秀毕业论文).

• Zepei Liu：推免研究生. He became a master student of Prof. Alexander KURGANOV in Sep. 2022.

• Manting Peng：推免研究生. She won SUSTech outstanding bachelor thesis (本科毕业论文入选南科大优秀毕业论文).

• Mingrui Wang: He pursues his master degree in electronic information at Peking University（北京大学）.

• Linfeng Xu：推免研究生. He won SUSTech outstanding bachelor thesis (本科毕业论文入选南科大优秀毕业论文).

• Luowei Yin：He worked on summer research and his bachelor thesis in our group (2021.04-2022.06) and after graduation pursues his Ph.D. at CUHK（香港中文）since Sep. 2022.

• Zijun Jia：推免研究生. 

• Yuchen Huang：推免研究生. He won SUSTech outstanding bachelor thesis (本科毕业论文入选南科大优秀毕业论文).

• Yuanji Zhong：He was selected into a joint study program in University of Wisconsin-Madison（威斯康星大学麦迪逊分校）. He won 1st class prize (Guangdong) and 3rd class prize (National) in the 13th Chinese Mathematics Competitions for University Students（全国大学生数学竞赛）. He won Silver Award in the 2nd Winter National Mathematical Olympiad for University Students（全国大学生奥林匹克数学竞赛-冬季赛）in 2023.

• Zhihua Li：He won SUSTech outstanding bachelor thesis (本科毕业论文入选南科大优秀毕业论文). He pursues his Ph.D. at University of Iowa（爱荷华大学）since Sep. 2023.

• Miaosheng Jiao：推免研究生. He won SUSTech outstanding bachelor thesis (本科毕业论文入选南科大优秀毕业论文).

Visiting Postdocs（访问博士后）

• Dr. Hao Li (2023.12-2024.1): Ph.D. from Purdue University；Postdoc at University of Texas at Austin. 

• Dr. Junming Duan (2023.6-2023.7): Ph.D. Peking University; Postdoc at EPFL. 

• Dr. Caiyou Yuan (2024.7-2024.8): Ph.D. Peking University.

Welcome passionate and highly self-motivated students to join our group! If you are interested in our research, please feel free to contact me by email: [email protected].

2 extra postdoc/RAP(research assistant professor) positions available. The candidates should have a Ph.D. degree in Mathematics, Computational Physics, Fluid Mechanics, Computer Science, or Astrophysics. Research experience in numerical schemes for PDEs, CFD, machine learning, and/or data science is desirable. Salary package is competitive. If you are interested, please send your CV to [email protected].

Publications List  

[68] S. Ding, K. Wu*, and C. Yuan

Divergence-free finite volume WENO scheme for relativistic magnetohydrodynamics preserving positivity and subluminal velocity

preprint,  2024.

[67] D. Pang and K. Wu*

A provably positivity-preserving constrained transport second-order scheme for ideal MHD equations

submitted,  2024.

[66] H. Cao, M. Peng, and K. Wu*

Robust discontinuous Galerkin methods maintaining physical constraints for general relativistic hydrodynamics

submitted,  2024.

[65] S. Cui, K. Wu*, and L. Xu

On local minimum entropy principle of high-order schemes for relativistic Euler equations

preprint,  2024.

[64] M. Peng, K. Wu*, and C. Yuan

Oscillation-eliminating central DG schemes for hyperbolic conservation laws

preprint, 2024.

[63] S. Ding, S. Cui, and K. Wu*
Robust DG schemes on unstructured triangular meshes: Oscillation elimination and bound preservation via optimal convex decomposition

submitted, 2024.

[62] Z. Zhang, H. Tang, and K. Wu*

High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: Well balancedness and positivity

submitted,  2024.

[61] J. Chen, K. Wu*, and D. Xiu

DUE: A deep learning framework and library for modeling unknown equations

submitted, 2024.

[60] Z. Li and K. Wu*

Spectral volume from a DG perspective: oscillation elimination, stability, and optimal error estimates 

submitted, 2024. 

[59] M. Liu and K. Wu*

Structure-preserving oscillation-eliminating discontinuous Galerkin schemes for ideal MHD equations: Locally divergence-free and positivity-preserving

submitted to JCP, 2024.

[58] L. Y. Win, Y. Chen, Y. Nie, L. Cai, and K. Wu

Physical-constraint-preserving high-order finite volume WENO method via geometric quasilinearization for Chaplygin gas dynamics
submitted, 2024.

[57] C. Cai, J. Qiu, and K. Wu*

Provably convergent and robust Newton-Raphson method: A new dawn in primitive variable recovery for relativistic MHD

submitted to SIAM Journal on Numerical Analysis, 2024.

[56] W. Chen, S. Cui, K. Wu, and T. Xiong

Bound-preserving OEDG method for Aw-Rascle-Zhang traffic models on networks

Journal of Computational Physics,   accepted 2024.

[55] J. Wang, H. Tang, K. Wu*

High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms

Journal of Computational Physics,   accepted 2024.

[54] C. Fan and K. Wu*

High-order oscillation-eliminating Hermite WENO method for hyperbolic conservation laws

Journal of Computational Physics,   accepted,  2024. 

[53] S. Ding and K. Wu*

GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics

Journal of Computational Physics,   accepted,  2024.          A special issue in honor of Sergei Godunov（俄罗斯科学院院士）

[52] S. Cui, A. Kurganov, and K. Wu*

Bound-preserving framework for central-upwind schemes for general hyperbolic conservation laws

SIAM Journal on Scientific Computing,   accepted, 2024.

[51] R. Yan, R. Abgrall, and K. Wu*

Uniformly high-order bound-preserving OEDG schemes for two-phase flow

Math. Models Methods Appl. Sci. (M3AS),    2024. 

[50] M. Peng, Z. Sun, and K.Wu*

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

Mathematics of Computation,   accepted, 2024

[49] J. Chen and K. Wu*

Positional knowledge is all you need: Position-induced Transformer (PiT) for operator learning

International Conference on Machine Learning (ICML),   accepted,  2024.

[48] W. Chen, K. Wu, and T. Xiong

High order structure-preserving finite difference WENO scheme for MHD equations with gravitation in all sonic Mach numbers 

Journal of Scientific Computing,  99: 36, 2024. 

[47] A. Chertock, A. Kurganov, M. Redle, and K. Wu

A new locally divergence-free path-conservative central-upwind scheme for ideal and shallow water magnetohydrodynamics

SIAM Journal on Scientific Computing,  accepted, 2024.

[46] C. Cai, J. Qiu, and K.Wu*

Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics

Journal of Computational Physics,   498: 112669, 2024.

[45] C. Zhang, K. Wu, and Z. He

Critical sampling for robust evolution operator learning of unknown dynamical systems

IEEE Transactions on Artificial Intelligence,   in press, 2024.

[44] L. Xu, S. Ding, and K. Wu*

High-order accurate entropy stable schemes for relativistic hydrodynamics with a general equation of state

Journal of Scientific Computing,  98: 43, 2024. 

[43] S. Ding and K.Wu*

A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations

SIAM Journal on Scientific Computing,   46: A50-A79, 2024.

[42] S. Cui, S. Ding, and K. Wu*

On optimal cell average decomposition for high-order bound-preserving schemes of hyperbolic conservation laws

SIAM Journal on Numerical Analysis,   62: 775-810, 2024. 

[41] J. Chen and K. Wu*

Deep-OSG: Deep learning of operators in semigroup

Journal of Computational Physics,   493: 112498, 2023.

[40] Y. Ren, K. Wu, J. Qiu, and Y. Xing

On high-order positivity-preserving well-balanced finite volume methods for the Euler equations with gravitation

Journal of Computational Physics,   492:112429, 2023.

[39] W. Chen, K. Wu, and T. Xiong

High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers
Journal of Computational Physics,  488:112240, 2023.

[38] S. Cui, S. Ding, and K. Wu*
Is the classic convex decomposition optimal for bound-preserving schemes in multiple dimensions?
Journal of Computational Physics,   476: 111882, 2023.

[37] K. Wu*, H. Jiang, and C.-W. Shu

Provably positive central DG schemes via geometric quasilinearization for ideal MHD equations
SIAM Journal on Numerical Analysis,   61: 250-285, 2023.

[36] K. Wu and C.-W. Shu
Geometric quasilinearization framework for analysis and design of bound-preserving schemes
SIAM Review,   (Research Spotlight)  65(4): 1031--1073, 2023.  

    This paper proposes a general approach---Geometric Quasi-Linearization (GQL), motivated by our previous bound-preserving works on MHD and RHD systems.  GQL equivalently transforms nonlinear constraints into linear ones, through properly introducing free auxiliary variables, i.e., it uses extra auxiliary variables in exchange for linearity.

[35] Z. Sun, Y. Wei, and K. Wu*

On energy laws and stability of Runge--Kutta methods for linear seminegative problems

SIAM Journal on Numerical Analysis,    60(5): 2448--2481, 2022.

[34] K. Wu 
Minimum principle on specific entropy and high-order accurate invariant region preserving numerical methods for relativistic hydrodynamics

SIAM Journal on Scientific Computing,   43(6): B1164--B1197, 2021.

[33] Z. Chen, V. Churchill, K. Wu, and D. Xiu
Deep neural network modeling of unknown partial differential equations in nodal space
Journal of Computational Physics,   449: 110782, 2022.

[32] K. Wu* and C.-W. Shu

Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations

Numerische Mathematik,  148: 699--741, 2021.

[31] Y. Chen and K. Wu*
A physical-constraint-preserving finite volume WENO method for special relativistic hydrodynamics on unstructured meshes
Journal of Computational Physics,   466: 111398, 2022.

[30] H. Jiang, H. Tang, and K. Wu*

Positivity-preserving well-balanced central discontinuous Galerkin schemes for the Euler equations under gravitational fields

Journal of Computational Physics,   463: 111297, 2022.

[29] K. Wu and Y. Xing

Uniformly high-order structure-preserving discontinuous Galerkin methods for Euler equations with gravitation: Positivity and well-balancedness

SIAM Journal on Scientific Computing,  43(1): A472--A510, 2021.

[28] K. Wu and D. Xiu

Data-driven deep learning of partial differential equations in modal space

Journal of Computational Physics,  408: 109307, 2020. 

[27] K. Wu, T. Qin, and D. Xiu

Structure-preserving method for reconstructing unknown Hamiltonian systems from trajectory data

SIAM Journal on Scientific Computing,  42(6): A3704--A3729, 2020. 

[26] K. Wu and C.-W. Shu

Entropy symmetrization and high-order accurate entropy stable numerical schemes for relativistic MHD equations

SIAM Journal on Scientific Computing,  42(4): A2230--A2261, 2020. 

[25] Z. Chen, K. Wu, and D. Xiu

Methods to recover unknown processes in partial differential equations using data

Journal of Scientific Computing,  85:23, 2020. 

[24] K. Wu, D. Xiu, and X. Zhong

A WENO-based stochastic Galerkin scheme for ideal MHD equations with random inputs 

Communications in Computational Physics,  30: 423--447, 2021.

[23] J. Hou, T. Qin, K. Wu and D. Xiu

A non-intrusive correction algorithm for classification problems with corrupted data

Commun. Appl. Math. Comput., 3: 337--356, 2021.

[22] K. Wu* and C.-W. Shu

Provably positive high-order schemes for ideal magnetohydrodynamics: Analysis on general meshes

Numerische Mathematik,  142(4): 995--1047, 2019.

[21] T. Qin, K. Wu, and D. Xiu

Data driven governing equations approximation using deep neural networks

Journal of Computational Physics,  395: 620--635, 2019.

[20] K. Wu and D. Xiu

Numerical aspects for approximating governing equations using data

Journal of Computational Physics,  384: 200--221, 2019.

[19] K. Wu and D. Xiu

Sequential approximation of functions in Sobolev spaces using random samples
Commun. Appl. Math. Comput.,  1: 449--466, 2019.

[18] K. Wu and C.-W. Shu

A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics

SIAM Journal on Scientific Computing,  40(5):B1302--B1329, 2018.

[17] K. Wu

Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics
SIAM Journal on Numerical Analysis,  56(4):2124--2147, 2018.

[16] Y. Shin, K. Wu, and D. Xiu

Sequential function approximation with noisy data

Journal of Computational Physics,  371:363--381, 2018.

[15] K. Wu and D. Xiu

Sequential function approximation on arbitrarily distributed point sets

Journal of Computational Physics,  354:370--386, 2018.

[14] K. Wu and H. Tang

On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state

Z. Angew. Math. Phys.,  69:84(24pages), 2018.

[13] K. Wu and D. Xiu

An explicit neural network construction for piecewise constant function approximation

arXiv preprint arXiv:1808.07390, 2018.

[12] K. Wu, Y. Shin, and D. Xiu

A randomized tensor quadrature method for high dimensional polynomial approximation

SIAM Journal on Scientific Computing,  39(5):A1811--A1833, 2017. 

[11] K. Wu

Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics

Physical Review D,  95, 103001, 2017. 

[10] K. Wu, H. Tang, and D. Xiu

A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

Journal of Computational Physics,  345:224--244, 2017. 

[9] K. Wu and H. Tang

Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations

Math. Models Methods Appl. Sci. (M3AS),  27(10):1871--1928, 2017. 

[8] Y. Kuang, K. Wu, and H. Tang

Runge-Kutta discontinuous local evolution Galerkin methods for the shallow water equations on the cubed-sphere grid

Numer. Math. Theor. Meth. Appl.,  10(2):373--419, 2017. 

[7] K. Wu and H. Tang

Physical-constraint-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state

Astrophys. J. Suppl. Ser. (ApJS),  228(1):3(23pages), 2017. (2015 Impact Factor of ApJS: 11.257)

[6] K. Wu and H. Tang

A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics

SIAM Journal on Scientific Computing,  38(3):B458--B489, 2016. 

[5] K. Wu and H. Tang

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

Applied Mathematics and Mechanics,  37(11):1441--1466, 2016. 

[4] K. Wu and H. Tang

High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

Journal of Computational Physics,  298:539--564, 2015.

[3] K. Wu, Z. Yang, and H. Tang

A third-order accurate direct Eulerian GRP scheme for one-dimensional relativistic hydrodynamics

East Asian J. Appl. Math.,  4(2):95--131, 2014.

[2] K. Wu and H. Tang

Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics

Journal of Computational Physics,  256:277--307, 2014. 

[1] K. Wu, Z. Yang, and H. Tang

A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics

Journal of Computational Physics,  264:177--208, 2014.