# Selected Recent Talks

Overview of Reduced Basis Method for Kinetic Transport

Overview of complexity reduction for kinetic equations

Introduction to kinetic models and high dimensional scientific computing

# Journal Publications

Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics, v223 (2007), pp.398-415.

Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Mathematics of Computation, v77 (2008), pp.699-730.

Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, Discontinuous Galerkin solver for Boltzmann-Poisson transients, Journal of Computational Electronics, v7 (2008), pp.119-123.

Y. Cheng and C.-W. Shu, Superconvergence and time evolution of discontinuous Galerkin finite element solutions, Journal of Computational Physics, v227 (2008), pp.9612-9627.

Y. Cheng and C.-W. Shu, Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-diffusion equations, Computers and Structures, v87 (2009), pp.630-641.

Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for Boltzmann Poisson systems in nano devices, Computer Methods in Applied Mechanics and Engineering, v198 (2009), pp.3130-3150.

Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, v47 (2010), pp. 4044-4072.

O. Bokanowski, Y. Cheng and C.-W. Shu, A discontinuous Galerkin solver for front propagation, SIAM Journal on Scientific Computing, v33 (2011), pp.923-938.

Y. Cheng, I. M. Gamba and J. Proft, Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations, Mathematics of Computation, v81 (2012), pp. 153-190.

Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations, SEMA Journal, v54 (2011), pp.47-64.

Y. Cheng, I. M. Gamba and K. Ren, Recovering doping profiles in semiconductor devices with the Boltzmann-Poisson model, Journal of Computational Physics, v230 (2011), pp. 3391-3412.

Y. Cheng and I. M. Gamba, Numerical study of one-dimensional Vlasov-Poisson equations in the simulation for infinite homogeneous stellar systems, Communications in Nonlinear Science and Numerical Simulation (Special Issue dedicated to P. J. Morrison’s 60th Birthday), v17 (2011), pp. 2052-2061.

Y. Cheng, I. M. Gamba and P. J. Morrison, Study of conservation and recurrence of Runge-Kutta discontinuous Galerkin schemes for Vlasov-Poisson systems, Journal of Scientific Computing, v56 (2013), pp. 319-349.

O. Bokanowski, Y. Cheng and C.-W. Shu, A discontinuous Galerkin scheme for front propagation with obstacles, Numerische Mathematik, v126 (2014), pp. 1-31.

Y. Cheng, A. J. Christlieb and X. Zhong, Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampére system, Journal of Computational Physics, v256 (2014), pp. 630-655.

R. Alonso, J. Young and Y. Cheng, A particle interaction model for the simulation of biological, cross-linked fiber networks inspired from flocking theory, Cellular and Molecular Bioengineering, v7 (2014), pp. 58-72.

Y. Cheng, I. M. Gamba, F. Li and P. J. Morrison, Discontinuous Galerkin methods for the Vlasov-Maxwell equations, SIAM Journal on Numerical Analysis, v52 (2014), pp. 1017-1049.

Y. Cheng and Z. Wang, A new discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics, v268 (2014), pp.134-153.

Y. Cheng, A. J. Christlieb and X. Zhong, Energy-conserving discontinuous Galerkin methods for the Vlasov-Maxwell system, Journal of Computational Physics, v279 (2014), pp. 145-173.

Y. Cheng, A. J. Christlieb and X. Zhong, Energy-conserving numerical simulations of electron holes in two-species plasmas, European Physical Journal D, v69 (2015), 67.

Y. Cheng, A. J. Christlieb and X. Zhong, Numerical study of the two-species Vlasov-Ampére system: energy-conserving schemes and the current-driven ion-acoustic instability, Journal of Computational Physics, v288 (2015), pp.66-85.

O. Bokanowski, Y. Cheng and C.-W. Shu, Convergence of discontinuous Galerkin schemes for front propagation with obstacles, Mathematics of Computation, v85 (2016), pp.2131-2159.

Y. Cheng, C.-S. Chou, F. Li and Y. Xing, L2 stable discontinuous Galerkin methods for one-dimensional two-way wave equations, Mathematics of Computation, v86 (2017), pp.121-155.

Z. Wang, Q. Tang, W. Guo and Y. Cheng, Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations, Journal of Computational Physics, v314 (2016), pp. 244-263.

W. Guo and Y. Cheng, A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations, SIAM Journal on Scientific Computing, v38 (2016), pp. A3381-A3409.

Y. Cheng, A. J. Christlieb, W. Guo and B. Ong, An asymptotic preserving Maxwell solver resulting in Darwin limit of electrodynamics, Journal of Scientific Computing, v71 (2017), pp. 959-993.

Y. Liu, Y. Cheng and C.-W. Shu, A simple bound-preserving sweeping technique for conservative numerical approximations, Journal of Scientific Computing, v73 (2017), pp. 1028-1071.

J. Morales-Escalante, I. M. Gamba, Y. Cheng, A. Majorana, C.-W Shu and J. Chelikowsky, Discontinuous Galerkin deterministic solvers for a Boltzmann-Poisson model of hot electron transport by averaged empirical pseudopotential band structures, Computer Methods in Applied Mechanics and Engineering, v321 (2017), pp. 209-234.

W. Guo and Y. Cheng, An adaptive multiresolution discontinuous Galerkin method for time-dependent transport equations in multi-dimensions, SIAM Journal on Scientific Computing, v39 (2017), pp. A2962–A2992.

V. A. Bokil, Y. Cheng, Y. Jiang and F. Li, Energy stable discontinuous Galerkin methods for Maxwell’s equations in nonlinear optical media, Journal of Computational Physics, v350 (2017), pp. 420-452.

R. J. Alonso, V. Bagland, Y. Cheng and B. Lods, One-dimensional dissipative Boltzmann equation: measure solutions, cooling rate, and self-similar profile, SIAM Journal on Mathematical Analysis, v50 (2018), pp. 1278-1321.

V. A. Bokil, Y. Cheng, Y. Jiang, F. Li and P. Sakkaplangkul, High spatial order energy stable FDTD methods for Maxwell’s equations in nonlinear optical media, Journal of Scientific Computing, v77 (2018), pp. 330-371.

A. Chen, F. Li and Y. Cheng, An ultra-weak discontinuous Galerkin method for Schrodinger equation in one dimension, Journal of Scientific Computing, v78 (2019), pp. 772-815.

P. Fu, Y. Cheng, F. Li and Y. Xu, Discontinuous Galerkin methods with optimal $L^2$ accuracy for PDEs with high order spatial derivatives, Journal of Scientific Computing, v78 (2019), pp. 816-863.

Z. Tao, W. Guo and Y. Cheng, Sparse grid discontinuous Galerkin methods for the Vlasov-Maxwell system, Journal of Computational Physics, Journal of Computational Physics:X, v3 (2019), 100022.

Y. Liu, Y. Cheng, S. Chen and Y.-T. Zhang, Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations, Journal of Computational Physics, Journal of Computational Physics, v388 (2019), pp. 90-102.

Z. Tao, A. Chen, M. Zhang and Y. Cheng, Sparse grid central discontinuous Galerkin method for linear hyperbolic systems in high dimensions, SIAM Journal on Scientific Computing, SIAM Journal on Scientific Computing, v41 (2019), pp. A1626-1651.

Y. Jiang, P. Sakkaplangkul, V. A. Bokil, Y. Cheng and F. Li, Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell’s equations in linear Lorentz media, Journal of Computational Physics, Journal of Computational Physics, v394 (2019), pp. 100-135.

Z. Peng, V. Bokil, Y. Cheng and 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.

M. Jiao, Y. Cheng, Y. Liu and M. Zhang, Central discontinuous Galerkin methods for the generalized Korteweg-de Vries equations, Communications in Computational Physics, v28 (2020), No. 3, pp. 927-966.

D. Appeleo, V. Bokil, Y. Cheng and 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.

A. Chen, Y. Cheng, Y. Liu and M. Zhang, Superconvergence of ultra-weak discontinuous Galerkin methods for the linear Schrodinger equation in one dimension, Journal of Scientific Computing, v82 (2020), pp. 1-44.

Z. Peng, Y. Cheng, J.-M. Qiu and 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.

J. Huang and Y. Cheng, An adaptive multiresolution discontinuous Galerkin method with artificial viscosity for scalar hyperbolic conservation laws in multidimensions, SIAM Journal on Scientific Computing , v42 (2020), A2943–A2973.

Z. Tao, Y. Jiang and Y. Cheng, An adaptive high-order piecewise polynomial based sparse grid collocation method with applications, Journal of Computational Physics , v43 (2021), pp. 109770.

J. Huang, Y. Liu, W. Guo, Z. Tao and Y. Cheng, An adaptive multiresolution interior penalty discontinuous Galerkin method for wave equations in second order form, Journal of Scientific Computing, to appear.

Z. Tao, J. Huang, Y. Liu, W. Guo and Y. Cheng, An adaptive multiresolution ultra-weak discontinuous Galerkin method for nonlinear Schrodinger equations, Communications on Applied Mathematics and Computation (Special issue on discontinuous Galerkin methods), to appear.

Z. Peng, Y. Cheng, J.-M. Qiu and F. Li, Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property, SIAM Journal on Numerical Analysis, to appear.

W. Guo, J. Huang, Z. Tao and Y. Cheng, An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions, Journal of Computational Physics, Journal of Computational Physics, v436 (2021), pp. 110294.

M. Lyu, V. Bokil, Y. Cheng and F. Li, Energy stable nodal discontinuous Galerkin methods for nonlinear Maxwell’s equations in multi-dimensions, Journal of Scientific Computing, to appear.

J. Huang, Y. Liu, Y. Liu, Z. Tao and Y. Cheng, A class of adaptive multiresolution ultra-weak discontinuous Galerkin methods for some nonlinear dispersive wave equations, SIAM Journal on Scientific Computing, to appear.

J. Huang, Y. Cheng, A. J. Christlieb and L. Roberts, Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure, Journal of Computational Physics, to appear.

Z. Peng, Y. Chen, Y. Cheng and F. Li, A reduced basis method for radiative transfer equation, Journal of Scientific Computing (Special issue on model order reduction), to appear. Highlighted on NSF Math Institute website

M. Lyu, V. Bokil, Y. Cheng and 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, to appear.

J. Huang, Y. Cheng, A. J. Christlieb and L. F. Roberts, Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds, Journal of Scientific Computing, to appear.

J. Huang, Y. Cheng, A. Christlieb, L. Roberts and W.-A. Yong, Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures, SIAM Journal on Multiscale Modeling and Simulation, to appear.

A. Galindo-Olarte, J. Huang, J. Ryan and Y. Cheng, Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov-Maxwell equations, BIT Numerical Mathematics, 2023(63).

J. Huang, W. Guo and Y. Cheng, Adaptive sparse grid discontinuous Galerkin method: review and software implementation, Communications on Applied Mathematics and Computation, 2023.

Z. Peng, Y. Chen, Y. Cheng and F. Li, A micro-macro decomposed reduced basis method for the time-dependent radiative transfer equation, Multiscale Modeling and Simulation, to appear.

# Preprints

- D. Appelo and Y. Cheng, Robust implicit adaptive low rank time-stepping methods for matrix differential equations, 2024.

# Software

Github package for adaptive sparse grid/multiresolution DG method