My research focuses on the construction, analysis, and application of numerical methods to solve partial differential equations that arise in emerging fields, especially those governed by hyperbolic conservation laws and those involving moving or complicated geometries. Examples of my current projects are: (1) an inherently superconvergent hybrid-variable (HV) discretization framework for hyperbolic conservation laws, (2) the shifted boundary method (SBM) for solving compressible fluid dynamics governed by the Euler equations in the context of free-boundary problems and fluid-structure interactions, as well as related reduced-order modeling (ROM) problems, (3) embedded boundary computation of multi-material flows on moving grids (ALE-EBM), (4) mathematical and numerical modeling of the cellular dynamics describing virus infection and virotherapy in early stage tumors, and (5) mathematical and numerical investigation of meandering instability in fluid-like systems.
Xianyi Zeng
Assistant Professor
2012 Ph.D., Computational Mathematics, Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA
2010 M.S., Financial Mathematics, Department of Statistics, Stanford University, Stanford, CA, USA
2006 B.S., Mathematics, School of Mathematical Sciences, Peking University, Beijing, China
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Research Areas
Additional Interests
- Numerical methods for partial differential equations
- Hyperbolic conservation laws
- Mathematical biology
- Computational mechanics
- High-performance computing
Research Statement
Biography
I graduated from the School of Mathematical Sciences at Peking University in 2006 and then started graduate study in computational mathematics at Stanford University. After receiving the Ph.D. in 2012 (with the dissertation topic on high-order methods for fluid-structure interactions), I worked as a postdoctoral research associate at Duke University until 2016. My postdoctoral research focuses on developing numerical methods for highly transient problems in computational mechanics and multi-physics applications. In Fall 2016, I joined the Department of Mathematical Sciences at The University of Texas at El Paso (UTEP) and was a core faculty member of the doctoral Computational Science Program and Data Science Program. While my tenure promotion was approved by the University of Texas system in 2022, I resigned and joined the Department of Mathematics at Lehigh University to reunite with my family.
1. X. Zeng and M. M. Hasan, "On the stability of explicit finite difference methods for advection-diffusion equations". Numer. Meth. Part. D. E. 39 (2023) 421–446.
2. H. D. Curtsinger, X. Zeng, Z. Mather, M. Ballyk, T. A. Phan, B. Niu, J. Pu, M. Y. Bartee, J. P. Tian, E. Bartee, "High levels of extracellular potassium can delay myxoma virus replication by preventing release of virions from the endosomes". J. Virol. (2023) Ahead of Print.
3. M. M. Hasan and X. Zeng, "A central compact hybrid-variable method with spectral-like resolution: One-dimensional case". J. Comput. Appl. Math. 421 (2023) 114894.
4. X. Zeng, G. Stabile, E. N. Karatzas, G. Scovazzi, and G. Rozza, "Embedded domain reduced basis models for the shallow water hyperbolic equations with the shifted boundary method". Comput. Methods Appl. Mech. Eng. 398 (2022) 115143.
5. B. Niu, X. Zeng, T. A. Phan, F. Szulzewsky, S. Holte, E. C. Holland, and J. P. Tian "Mathematical modeling of PDGF-driven glioma reveals the dynamics of immune cells infiltrating into tumors". Neoplasia 22 (2020) 323–332.
6. X. Zeng, Mashriq A. Saleh, and Jianjun P. Tian, "On finite volume discretization for infiltration dynamics in tumor growth models". Adv. Comput. Math. 45 (2019) 3057–3094.
7. X. Zeng, "Linear hybrid-variable methods for advection equations". Adv. Comput. Math. 45 (2019) 929–980.
8. X. Zeng, K. Li, and G. Scovazzi, "An ALE/embedded boundary method for two-material flow simulations". Comput. Math. Appl. 78 (2019) 335–361.
9. X. Zeng, G. Scovazzi, N. Abboud, O. Colom ́es Gene, and S. Rossi, "A dynamic variational multiscale method for viscoelasticity using linear tetrahedral elements". Int. J. Num. Meth. Eng. 112 (2017) 1951–2003.
10. G. Scovazzi, T. Song, and X. Zeng, "A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strong and weak boundary conditions". Comput. Methods Appl. Mech. Eng. 325 (2017) 532–576.
11. A. Main, X. Zeng, P. Avery, and C. Farhat, "An enhanced FIVER method for multi-material flow problems with second-order convergence rate". J. Comput. Phys. 329 (2017) 141–172.
12. X. Zeng, "A General Approach to Enhance Slope Limiters in MUSCL Schemes on Non-Uniform Rectilinear Grids". SIAM J. Sci. Comput. 38 (2016) A789–A813.
13. X. Zeng and G. Scovazzi, "A variational multiscale finite element method for monolithic ALE computations of shock hydrodynamics using nodal elements". J. Comput. Phys. 315 (2016) 577–608.
14. G. Scovazzi, B. Carnes, X. Zeng, and S. Rossi, "A simple, stable and accurate tetrahedral finite element for transient, nearly incompressible, linear and nonlinear elasticity: A dynamic variational multiscale approach". Intl. J. Num. Meth. Eng. 106 (2016) 799–839.
15. X. Zeng and G. Scovazzi, "A frame-invariant vector limiter for flux corrected nodal remap in arbitrary Lagrangian-Eulerian flow computations". J. Comput. Phys. 270 (2014) 753–783.
16. X. Zeng, "A High-Order Hybrid Finite Difference-Finite Volume Approach with Application to Inviscid Compressible Flow Problems: A Preliminary Study". Comput. Fluids 98 (2014) 91–110.
17, X. Zeng and C. Farhat, "A Systematic Approach for Constructing Higher-Order Immersed Boundary and Ghost Fluid Methods for Fluid-Structure Interaction Problems". J. Comput. Phys. 231 (2012) 2892–2923.
Teaching
MATH 319 Spring 2023 Introduction to Differential Equations
In this course we will use several standard techniques to find analytical solutions to various ordinary differential equations (ODE) and partial differential equations (PDE), including both scalar ones and systems. In addition, we will perform qualitative analysis of scalar ODEs in a more general setting and get exposed to uniqueness and existence theory of these equations. Specific topics include first-order scalar ODEs and ODE systems, second-order linear ODEs, comparison principle and well-posedness of first-order ODEs, and introduction to linear and nonlinear PDEs.