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, 2022.

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 DG schemes via geometric quasilinearization for ideal MHD equations, SIAM Journal on Numerical Analysis, 2022.

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.
S. Cui, S. Ding, and K. Wu*, On optimal cell average decomposition for high-order bound-preserving schemes of hyperbolic conservation laws, preprint, 2022. 
Y. Ren, K. Wu, J. Qiu, and Y. Xing, On positivity-preserving well-balanced finite volume methods for the Euler equations with gravitation, preprint, 2022. 

S. Ding and K. Wu*, A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations, preprint, 2023.

Deep Learning and Data-driven Modeling

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.

J. Chen and K. Wu*, Deep-OSG: A deep learning approach for approximating a family of operators in semigroup to model unknown autonomous systems, preprint, 2022. 

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.

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

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）

Group Members

Research Assistant Professor

 ◆  Dr. Shumo CUI (2023.2.1-) 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" submitted to SINUM. 

Postdoctoral Fellows
  ◆  Dr. Shengrong DING (2021.11-present):  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" submitted to SINUM. We have a joint article "A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations" submitted to SISC.

  ◆  Dr. Junfeng CHEN (Postdoctoral Fellow: 2022.10-present; Visiting Postdoc Scholar: 2022.03-2022.09):  B.Sc. from Tsinghua University（清华本科），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: A deep learning approach for approximating a family of operators in semigroup to model unknown autonomous systems" submitted to JCP.

Graduate Students

  ◆  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" accepted for publication in《SIAM Journal on Numerical Analysis》. We have a joint article with Prof. Huazhong Tang: "Positivity-preserving well-balanced central discontinuous Galekin schemes for the Euler equations under gravitational fields" published in《Journal of Computational Physics》.  

  ◆  Fang YAN (2021.09-)，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）.

  ◆  Manting PENG (2022.09-)，Master Student，B.Sc. from SUSTech（南科大）.
  ◆  Linfeng XU (2022.09-)，Master Student，B.Sc. from SUSTech（南科大）.

Undergraduate Students

  ◆  Xinran FANG 

  ◆  Yunhao JIANG：He was selected into a joint study program in University of Wisconsin-Madison. 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：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：He became a master student of Prof. Alexander KURGANOV in Sep. 2022. 

  ◆  Manting PENG：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 

  ◆  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 

  ◆  Linfeng XU：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

  ◆  Yuanji ZHONG

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: WUKL@sustech.edu.cn.

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 WUKL@sustech.edu.cn.

Publications List  

[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, submitted, 2023.

[47] 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, submitted, 2023.

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

Critical sampling for robust evolution operator learning of unknown dynamical systems, submitted, 2023.

[45] L. Xu, S. Ding, and K. Wu*

High-order accurate entropy stable schemes for relativistic hydrodynamics with a general equation of state, submitted, 2023. 

[44] S. Ding and K.Wu*

A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations, submitted, 2023. 

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

On optimal cell average decomposition for high-order bound-preserving schemes of hyperbolic conservation laws, submitted, 2023. 

[42] J. Chen and K. Wu*

Deep-OSG: A deep learning approach for approximating a family of operators in semigroup to model unknown autonomous systems, submitted, 2023. 

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

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

Journal of Computational Physics,  submitted, 2022.

[40] 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,  submitted, 2022. 

[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,  accepted, 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,   accepted, 2022.

[36] 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.

[35] K. Wu and C.-W. Shu*
Geometric quasilinearization framework for analysis and design of bound-preserving schemes
SIAM Review,    accepted, 2022.  arXiv:2111.04722.  8 Nov 2021

[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.