ReInterests

Editorship

Editorial Board Member, Journal of Scientific Computing. 2014 - Present.
Associate Editor, Communications on Applied Mathematics and Computation. 2019 - Present.
Editorial Board Member - Electronic Research Archive. AIMS Press. 2024 - Present.
Associate Editor, Frontiers in Applied Mathematics and Statistics. 2023 - Present.
Editorial Board Member - Mathematics. MDPI. 2023 - Present.
Guest Editor, Pure and Applied Mathematics Quarterly. 2016 - 2018.
Guest Editor, Beijing Journal of Pure and Applied Mathematics. 2024 - 2025.

Current Research Interests

Numerical Methods and Scientific Computing: researching high-order accuracy numerical methods for PDEs, including weighted ENO finite volume / finite difference methods, discontinuous Galerkin (DG) finite element methods, large time-step-size schemes (e.g., implicit integration factor schemes and their high dimensional implementation, exponential time differencing schemes, and other exponential integrators), fast and robust iterative methods (e.g., fast sweeping methods for steady state problems), application of machine learning methods to numerical PDEs, and high-order accuracy numerical methods on unstructured meshes for complex domain geometries, etc.

Applications: numerical solution of PDEs arising in compressible flow problems in fluid dynamics, wave problems in physics, and morphogenesis / tissue patterning problems in biology; computational analysis of physical and biological problems.

Group members (recent and former)

o Dr. Rentian Hu, Ph.D., May 2025. Thesis title: ABSOLUTELY CONVERGENT FAST SWEEPING WENO METHODS FOR EIKONAL EQUATIONS AND HYBRID WENO3 METHOD WITH MLP INDICATOR FOR CONSERVATION LAWS.

o Dr. Ernie Tsybulnik, Ph.D., May 2023. Thesis title: EFFICIENT SPARSE-GRID IMPLEMENTATION OF A FIFTH-ORDER MULTI-RESOLUTION WENO SCHEME FOR HYPERBOLIC EQUATIONS.

o Dr. Zachary Miksis, Ph.D., August 2022. Thesis title: Sparse-grid implementation of fixed-point fast sweeping WENO schemes for Eikonal equations.

o Dr. Xiaozhi Zhu, Ph.D., August 2021. Thesis title: FAST SPARSE GRID SIMULATIONS OF FIFTH ORDER WENO SCHEME FOR HIGH DIMENSIONAL HYPERBOLIC PDES".

o Dr. Dong Lu, Ph.D., August 2017. Thesis title: KRYLOV INTEGRATION FACTOR METHOD FOR HIGH SPATIAL DIMENSION CONVECTION-DIFFUSION PROBLEMS ON SPARSE GRIDS.

o Dr. Michael Machen, Ph.D., May 2017. Thesis title: KRYLOV IMPLICIT INTEGRATION FACTOR METHODS FOR SOLVING FOURTH ORDER EQUATIONS.

o Dr. Liang Wu, Ph.D., May 2016. Thesis title: High Order Fast Iterative Methods for Steady State of Hyperbolic Partial Differential Equations.

o Dr. Tian Jiang, Ph.D., May 2015. Thesis title: Krylov Implicit Integration Factor WENO Methods for Stiff Advection-Diffusion-Reaction Equations.

o Dr. Yuan Liu, Ph.D., August 2012. Thesis title: WENO Computations and Pattern Stability.

o Dr. Jianfeng Zhu, Ph.D., May 2010. Thesis title: Application of Discontinuous Galerkin Finite Element Methods for Vertebrate Limb Pattern Formation.

o Postdoc fellow and Project

o Dr. Ziyao Xu (Ph.D., Brown University), July 2023 - August 2025 (postdoc). Project title: high-order exponential time differencing methods with WENO and discontinuous Galerkin spatial discretizations for nonlinear degenerate parabolic PDEs.

ACMS Colloquium: 2011-2012

Notre Dame Conferences:
Midwest Numerical Analysis Days 2012 - Website