Fengyan Li

Mathematical Sciences
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About

Dr. Fengyan Li received her BS and MS degrees in Computational Mathematics from Peking University in 1997 and 2000, respectively, and her PhD degree in Applied Mathematics from Brown University in 2004. Before she joined RPI in 2006, she held a postdoc position at University of South Carolina.

Dr. Li's research interests and activities are mainly in Numerical Analysis and Scientific Computing. Her research focuses on the design and analysis of robust and highly accurate computational methods, especially discontinuous Galerkin finite element methods, with applications such as in wave propagation, fluid dynamics, rarefied gas dynamics, plasma physics, astrophysics, and nonlinear optics. Her recent research activities are mostly on multi-scale simulations and reduced order modelings for kinetic transport problems. Dr. Li received a Stella Dafermos Award at Brown University in 2004, and was a recipient of Alfred P. Sloan Research Fellowship in 2008. In 2009, she was granted an NSF-CAREER award for her research in high order methods and their applications. Dr. Li. was a plenary speaker at the 2015 Annual Meeting of Computational Mathematics in China (Guangzhou, China) , at the International Conference on Spectral and High Order Methods (ICOSAHOM, London, UK) in 2018, and at the North American High Order Methods Conference (NAHOMCon, San Diego, USA) in 2022. Currently, Dr. Li is serving on the editorial board of IMA Journal of Numerical Analysis, Mathematics of Computation, SIAM Journal of Numerical Analysis, CSIAM Transaction on Applied Mathematics, and Applied Mathematics and Mechanics (English Edition). She was an associate editor for SIAM Journal of Numerical Computing during 2014-2019.

Dr. Li has been actively involved in Association for Women in Mathematics (AWM) and Women in Numerical Analysis and Scientific Computing (WINASC) Research Network. In 2015, she co-organized a mini-symposium and served as a career panelist at the 8th International Congress on Industrial and Applied Mathematics (ICIAM) in Beijing (China), she was also an invited speaker at the workshop, Women in Applied Maths & Soft Matter Physics, in Mainz (Germany); She was a mentor in the AWM Mentor Network for more than a decade, and has been a faculty advisor of the AWM Student Chapter at RPI since 2016. Dr. Li is currently a member of the Steering Committee of WINASc. Recently, Dr. Li was elected as a 2025 Class of AWM Fellow.

Education & Training
  • Ph.D. in Applied Mathematics, Brown University, 2004
  • M.S. in Computational Mathematics, Peking University, 2000
  • B.S. in Computational Mathematics, Peking University, 1997

Research

General interests: Design, analysis, implementation and applications of accurate, robust and efficient numerical methods for PDEs

  • Discontinuous Galerkin methods; Finite element methods
  • High order methods for MHD, Hamilton-Jacobi, Maxwell's equations in linear and nonlinear media, Vlasov-Maxwell equations, multi-scale kinetic transport models, fluid models
  • High order structure-preserving methods: divergence-free, well-balanced, energy stable/conservative, positivity preserving, asymptotic preserving
  • Model order reduction techniques: learning based algorithms for transport dominated models, reduced order models for radiative transfer simulation 

Recognition

Awards & Honors
  • 2025 Class of AWM (Association for Women in Mathematics) Fellow
  • National Science Foundation (NSF) CAREER Award, 2009
  • Alfred P. Sloan Research Fellow, 2008
  • Stella Dafermos Award, Brown University, 2004
  • Ostrach Fellowship, Brown University, 2003 - 2004
Presentations & Appearances
  • Plenary / Conference Invited Speaker
    • The first Annual Meeting of the New York-New Jersey-Pennsylvania Section of the Society for Industrial and Applied Mathematics (SIAM-NNP), New Jersey Institute of Technology, Newark, Oct 20-22, 2023
    • The 2nd North American High Order Methods Conference (NAHOMCon), San Diego State University, San Diego, July 18-19, 2022
    • International Conference on Spectral and High Order Methods (ICOSAHOM), Imperial College London, London, United Kingdom, July 9-13, 2018
    • XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Aachen, Germany, August 1-5, 2016
    • The Annual Meeting of Computational Mathematics in China, Guangzhou, China, September 19-22, 2015

Publications

Google Scholar

  • In Refereed Journals (submitted, accepted or published)
    • 56) M. Lyu, F. Li, Nodal discontinuous Galerkin methods for Maxwell’s equations in Lorentz-Kerr-Raman medium without nonlinear algebraic solver, Journal of Computational Physics, v515 (2024), pp.113280
    • 55) Z. Peng, Y. Chen, Y. Cheng, F. Li, A micro-macro decomposed reduced basis method for the time-dependent radiative transfer equation, [PDF], Multiscale Modeling & Simulation, v22 (2024), pp.639-666
    • 54) M. Lyu, V.A. Bokil, Y. Cheng, F. Li, Energy stable nodal DG methods for Maxwell's equations of mixed-order form in nonlinear optical media, Communications on Applied Mathematics and Computation, v6 (2024)
    • 53) H. Wang , F. Li, C.-W. Shu, Q. Zhang, Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation, Mathematics of Computation, v 92 (2023), 2475-2513
    • 52) D. Appelo, L. Zhang, T Hagstrom, F. Li, An energy-based discontinuous Galerkin method with tame CFL numbers for the wave equation, [PDF], BIT Numerical Mathematics, v63 (2023)
    • 51) Z. Peng, M. Wang, F. Li, A learning-based projection method for model order reduction of transport problems [PDF], Journal of Computational and Applied Mathematics, v418 (2023), doi.org/10.1016/j.cam.2022.114560
    • 50) Z. Peng, Y. Chen, Y. Cheng, F. Li, A reduced basis method for radiative transfer equation, [PDF], Journal of Scientific Computing, v91 (2022), doi.org/10.1007/s10915-022-01782-2 (Special issue dedicated to the ICERM Spring 2020 semester program on Model Order Reduction)
    • 49) M. Lyu, V.A. Bokil, Y. Cheng, F. Li, Energy stable nodal discontinuous Galerkin methods for nonlinear Maxwell's equations in multi-dimensions, Journal of Scientific Computing, v89 (2021), https://doi.org/10.1007/s10915-021-01651-4
    • 48) Z. Peng and F. Li, Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on Schur complement, [PDF], SIAM Journal on Scientific Computing, v43 (2021), pp.A1194-A1220
    • 47) Z. Peng, Y. Cheng, J.-M. Qiu, F. Li, Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property, [PDF], SIAM Journal on Numerical Analysis, v59 (2021), pp.925-954
    • 46) Z. Peng, Y. Cheng, J.-M. Qiu, F. Li, Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling, Journal of Computational Physics, v415 (2020), pp.109485
    • 45) Z. Peng, V.A. Bokil, Y. Cheng, F. Li, Asymptotic and positivity preserving methods for Kerr-Debye model with Lorentz dispersion in one dimension, Journal of Computational Physics, v402 (2020), pp.109101
    • 44) D. Appelo, V.A. Bokil, Y. Cheng, F. Li, Energy stable SBP-FDTD methods for Maxwell-Duffing models in nonlinear photonics, IEEE Journal on Multiscale and Multiphysics Computational Techniques, v4 (2019), pp.329-336
    • 43) Y. Jiang, P. Sakkaplangkul, V.A. Bokil, Y. Cheng, F. Li, Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media, [PDF], Journal of Computational Physics, v394 (2019), pp.100-135
    • 42) A. Chen, F. Li, Y. Cheng, An ultra-weak discontinuous Galerkin method for Schrodinger equation in one dimension, [PDF], Journal of Scientific Computing, v78 (2019), pp.772-815
    • 41) P. Fu, Y. Cheng, F. Li, Y Xu, Discontinuous Galerkin methods with optimal L2 accuracy for one dimensional linear PDEs with high order spatial derivatives, Journal of Scientific Computing, v78 (2019), pp.816-863
    • 40) V.A. Bokil, Y. Cheng, Y. Jiang, F. Li, P. Sakkaplangkul, High spatial order energy stable FDTD methods for Maxwell's equations in nonlinear optical media in one dimension, Journal of Scientific Computing, v77 (2018), pp.330-371
    • 39) P. Fu, F. Li, Y. Xu, Globally divergence-free discontinuous Galerkin methods for ideal Magnetohydrodynamic equations, Journal of Scientific Computing, v77 (2018), pp.1621-1659
    • 38) V. A. Bokil, Y. Cheng, Y. Jiang, F. Li, Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media, [PDF], Journal of Computational Physics, v350 (2017), pp.420-452
    • 37) H. Yang and F. Li, Discontinuous Galerkin methods for relativistic Vlasov-Maxwell system, Journal of Scientific Computing, v73 (2017), pp.1216-1248
    • 36) M. Li, P. Guyenne, F. Li, L. Xu, A positivity-preserving well-balanced central discontinuous Galerkin method for the nonlinear shallow water equations, Journal of Scientific Computing, v71 (2017), pp.994-1034
    • 35) Y. Cheng, C.-S. Chou, F. Li, Y. Xing, L2 stable discontinuous Galerkin methods for one-dimensional two-way wave equations, Mathematics of Computation, v86 (2017), pp.121-155.
    • 34) M. Li, F. Li, Z. Li, L. Xu, Maximum-principle-satisfying and positivity-preserving high order central DG methods for hyperbolic conservation laws, SIAM Journal on Scientific Computing, v38 (2016), pp.A3720-A3740
    • 33) Z. Tao, F. Li, J. Qiu, High-order central Hermite WENO schemes: dimension-by-dimension moment-based reconstructions, Journal of Computational Physics, v318 (2016), pp.222-251
    • 32) F. Long, F. Li, X. Intes, S.P. Kotha, Radiative transfer equation modeling by streamline diffusion modified continuous Galerkin method, The Journal of Biomedical Optics, v21 (2016), 036003
    • 31) H. Yang and F. Li, Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations, ESAIM: Mathematical Modelling and Numerical Analysis, v50 (2016), pp.965-993
    • 30) Z. Tao, F. Li, J. Qiu, High-order central Hermite WENO schemes for hyperbolic conservation laws, Journal of Computational Physics, v281 (2015), pp.148-176
    • 29) T. Xiong, J. Jang, F. Li, J.-M. Qiu, High order asymptotic preserving nodal discontinuous Galerkin IMEX schemes for the BGK equation, [PDF], Journal of Computational Physics, v284 (2015), pp.70-94
    • 27) M.A. Reyna and F. Li, Operator bounds and time step conditions for DG and central DG methods, Journal of Scientific Computing, v62 (2015), pp.532-554 (Note: correction is made to definition 9 on page 10)
    • 26) J. Gopalakrishnan, F. Li, N.-C. Nguyen, J. Peraire, Spectral approximations by the HDG method,Mathematics of Computation, v84 (2015), pp.1037-1059.
    • 25) H. Yang and F. Li, Error estimates of Runge-Kutta discontinuous Galerkin methods for the Vlasov-Maxwell system, [PDF], ESAIM: Mathematical Modelling and Numerical Analysis, v49 (2015), pp.69-99.
    • 24) J. Jang, F. Li, J.-M. Qiu, T. Xiong, Analysis of asymptotic preserving DG-IMEX schemes for linear kinetic transport equations in a diffusive scaling, [PDF], SIAM Journal on Numerical Analysis, v52 (2014), pp. 1497-2206
    • 23) M. Li, P. Guyenne, F. Li, L. Xu, High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model, Journal of Computational Physics, v257 (2014), pp.169-192
    • 22) Y(ingda). Cheng, I. Gamba, F. Li, P. Morrison, Discontinuous Galerkin methods for Vlasov-Maxwell equations, [PDF],SIAM Journal on Numerical Analysis, v52-2 (2014), pp. 1017-1049 (see arxiv:1302.2136 for a long version)
    • 21) H. Yang, F. Li, J. Qiu, Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods, Journal of Scientific Computing, v55 (2013), pp.552-574
    • 20) Y(ue). Cheng, F. Li, J. Qiu, L. Xu, Positivity-preserving DG and central DG methods for ideal MHD equations, Journal of Computational Physics, v238 (2013), pp.255-280
    • 19) J. Jang, F. Li, J.-M. Qiu, T. Xiong, High Order Asymptotic Preserving DG-IMEX Schemes for Discrete-Velocity Kinetic Equations in a Diffusive Scaling , [PDF], Journal of Computational Physics, v281 (2015), pp. 199-224
    • 18) F. Li and L. Xu, Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations, Journal of Computational Physics, v231 (2012), pp.2655-2675
    • 17) F. Li, On the negative-order norm accuracy of a local-structure-preserving LDG method, Journal of Scientific Computing,v51 (2012), pp.213-223
    • 16) F. Li, L. Xu, S. Yakovlev, Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, Journal of Computational Physics, v230 (2011), pp.4828-4847. (Correction: in section 4.2.6, it is `with the darker area representing the smaller value)
    • 15) W. Guo, F. Li, J. Qiu, Local-structure-preserving discontinuous Galerkin methods with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations, Journal of Scientific Computing, v47 (2011), pp.239-257
    • 14) Y.-T. Zhang, S. Chen, F. Li, H.-K. Zhao, C.-W. Shu, Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations, SIAM Journal on Scientific Computing, v33 (2011), pp.1873-1896
    • 13) B. Cockburn, J. Gopalakrishnan, F. Li, N.-C. Nguyen, J. Peraire, Hybridization and postprocessing techniques for mixed eigenfunctions, SIAM Journal on Numerical Analysis, v48 (2010), pp.857-881
    • 12) F. Li and S. Yakovlev, A central discontinuous Galerkin method for Hamilton-Jacobi equations, Journal of Scientific Computing, v45 (2010), pp.404-428. (Special issue in memory of Professor David Gottlieb)
    • 11) S. C. Brenner, F. Li, L.-Y. Sung, Nonconforming Maxwell eigensolvers, Journal of Scientific Computing, v40 (2009), pp.51-85
    • 10) S. C. Brenner, F. Li, L.-Y. Sung, A nonconforming penalty method for a two-dimensional curl-curl problem, Mathematical Models and Methods in Applied Mathematics, v19 (2009), pp.651-668
    • 9) F. Li, C.-W. Shu, Y.-T. Zhang, H.-K. Zhao, A second order DGM based fast sweeping method for Eikonal equations, Journal of Computational Physics, v227 (2008), pp.8191-8208
    • 8) S. C. Brenner, J. Cui, F. Li, L.-Y. Sung, A nonconforming finite element method for a two-dimensional curl-curl and grad-div problem, Numerische Mathematik, v109 (2008), pp.509-533
    • 7) S. C. Brenner, F. Li, L.-Y. Sung, A locally divergence-free interior penalty method for two-dimensional curl-curl problems, SIAM Journal on Numerical Analysis, v46 (2008), pp.1190-1211
    • 6) S. C. Brenner, F. Li, L.-Y. Sung, A locally divergence-free nonconforming finite element method for the reduced time-harmonic Maxwell equations, Mathematics of Computation, v76 (2007), pp.573-595
    • 5) F. Li and C.-W. Shu, A local-structure-preserving local discontinuous Galerkin method for the Laplace equation,[some correction] Methods and Applications of Analysis, v13 (2006), pp.215-233 (Special issue dedicated to Professor Bjorn Engquist on the occasion of his 60th birthday)
    • 4) F. Li and C.-W. Shu, Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations, Applied Mathematics Letters, v18 (2005), pp.1204-1209
    • 3) F. Li and C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for MHD equations, Journal of Scientific Computing, v22-23 (2005), pp.413-442
    • 2) B. Cockburn, F. Li, C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, Journal of Computational Physics, v194 (2004), pp.588-610
    • 1) L.-A. Ying and F. Li, Exterior Problem of the Darwin Model and its Numerical Computation, ESAIM: Mathematical Modelling and Numerical Analysis, v37 (2003), pp.515-532
  • Refereed Book Chapters
    • B1) Y. Chen, Z. Chen, Y. Cheng, A. Gillman, and F. Li, Study of discrete scattering operators for some linear kinetic models, The IMA Volumes in Mathematics and its Applications, Vol. 160 (2016), Susanne Brenner (Ed): Topics in Numerical Partial Differential Equations and Scientific Computing, pp.99-136, Springer.
  • Publications in Conference and Workshop Proceedings
    • [2] S. C. Brenner, F. Li and L.-Y. Sung, A locally divergence-free nonconforming finite element method for the reduced time-harmonic Maxwell equations, Proceedings of the joint Workshop by AWM and MSRI: The Legacy of Ladyzhenskaya and Oleinik, 2006, pp.187-191
    • [1] B. Cockburn, F. Li and C.-W. Shu, Discontinuous Galerkin methods for equations with divergence-free solutions: preliminary results, the proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, K.J. Bathe, Editor, June 2003, Elsevier Science, pp.1900-1902
  • Research report
    • F. Li, A priori error estimates of a local-structure-preserving LDG method, May 2011

Research Group

Current Member

  • PhD Student: Kimberly Matsuda, Summer 2021 - present
  • We are always looking for motivated students to join the group.

Former Members

  • Postdoc:
    • David Wells, Fall 2015 - Summer 2018, Currently: UNC Chapel Hill
    • Liwei Xu, Fall 2009- Spring 2012, Currently: Professor at University of Electronic Science and Technology of China, China
  • PhD Students:
    • Zhichao Peng, Ph.D. 2020, First job: postdoc in Department of Mathematics, Michigan State University; Currently: Assistant Professor at Hong Kong University of Science and Technology
    • Matthew Reyna, Ph.D. 2014, First job: postdoc in Computer Science Department, Brown University; Currently: Assistant Professor in the Department of Biomedical Informatics & the Department of Pharmacology and Chemical Biology, Emory University
    • He Yang, Ph.D. 2014, First job: postdoc in Radiological Sciences Laboratory, Stanford University; Currently: Associate Professor in Department of Mathematics, Augusta University
    • Sergey Yakovlev, Ph.D. 2011, First job: postdoc in Scientific Computing and Imaging Institute, University of Utah; Currently: Senior Software Engineer, FD Software Enterprises, Inc
  • Visiting Doctoral/Master Students:
    • Maohui Lyu, August 2018-August 2019, from Chongqing University, Doctoral. First job: postdoc at LSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, China; Currently: Associate Professor at Beijing University of Posts and Telecommunications
    • Pei Fu, February-July 2015, February-June 2017, from University of Science and Technology of China, Doctoral. First job: postdoc at University of Science and Technology of China; Currently: Associate Professor at Nanjing University of Aeronautics & Astronautics, China 
    • Zhanjing Tao, January-July 2014, from Xiamen University, China, Doctoral. First job: postdoc in the Department of Mathematics, Michigan State University; Currently: Associate Professor at Jilin University
    • Maojun Li, August 2011-August 2012, from Chongqing University, Doctoral. First job: postdoc at Bejing Computational Science Research Center, China; Currently: Professor at University of Electronic Science and Technology of China, China 
    • Yue Cheng, Spring 2011, from Nanjing University, China, Master
  • Undergraduates (Recent):
    • Jingmin Sun, Spring 2020, Topic: On Sticky Brownian Motion and Numerical Solution; Currently a doctoral student at Carnegie Mellon University
    • Zachary J. Frangella, Summer 2018, Topic: Numerical Methods for Parabolic Equations with Non-Smooth Initial Data; Currently a doctoral student at Stanford University
    • Xiaoan Shen, Summer 2015, Summer and Fall 2016, Topic: Mathematical Characterization of Bound Preserving Implicit Schemes; Currently a doctoral student at University of Illinois, Chicago

Research Network WINASc, and AWM

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