XIANGXIONG ZHANG

Books:

Ubiquitous Laplacian: An Introduction to Numerical PDEs with Applications in Data Science, R. Lai and X. Zhang, World Scientific, 2026.

Preprints and publications:

Preprints

Publications in Refereed Book Chapters and Conference Proceedings

Publications and Accepted Papers in Refereed Journals:

56. K. Wu, X. Zhang and C.-W. Shu, High Order Numerical Methods Preserving Invariant Domain for Hyperbolic and Related Systems, to appear in SIAM Review. PDF arXiv

55. A. Anshika, J. Li, D. Ghosh and X. Zhang, A Three-Operator Splitting Scheme Derived from Three-Block ADMM, MOPTA 24 special issue of Optimization and Engineering (2025). DOI PDF arXiv code
54. C. Liu, J. Hu, W. Taitano and X. Zhang, An optimization-based positivity-preserving limiter in semi-implicit discontinuous Galerkin schemes solving Fokker–Planck equations, Computers and Mathematics with Applications, Vol. 192, 15 (2025), pp. 54-71. DOI PDF arXiv
53. C. Liu. Z. Sun and X. Zhang, A bound-preserving Runge--Kutta discontinuous Galerkin method with compact stencils for hyperbolic conservation laws, Journal of Computational Physics, Vol 537 (2025), 114071. arXiv
52. S. Zheng, W. Huang, B. Vandereycken and X. Zhang, Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints, Computational Optimization and Applications (2025) Volume 91, pages 1135–1184, DOI. PDF. An extended version with more details is on arXiv.

51. W. Hao, S. Lee and X. Zhang, An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems, Journal of Scientific Computing, (2025) 103:89. DOI arXiv
50. Y. Chen, D. Xiu and X. Zhang, On enforcing non-negativity in polynomial approximations in high dimensions, SIAM Journal on Scientific Computing, 47, no. 2 (2025), A866-A888. DOI PDF arXiv
49. B. Ren, B.S. Wang, X. Zhang and Z. Gao, Positivity and Bound Preserving Well-Balanced Compact Finite Difference Scheme for Ripa and Pollutant Transport Systems, Computers and Mathematics with Applications, Vol. 176, 15 (2024), pp. 545-563. PDF
48. Z. Chen, J. Lu, Y. Lu and X. Zhang, Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem, Mathematics of Computation 94 (2025), 2723-2760. DOI arXiv
47. C. Liu, G. Buzzard and X. Zhang, An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier–Stokes equations, Journal of Computational Physics, 519 (2024), pp. 113440. PDF arXiv
46. H. Li and X. Zhang, A monotone Q1 finite element method for anisotropic elliptic equations, the special issue in honor of Prof. Chi-Wang Shu’s 65th birthday in Beijing Journal of Pure and Applied Mathematics, Vol. 2, no. 1 (2025), pp. 183–217. DOI PDF arXiv
45. X. Liu, J. Shen and X. Zhang, A simple GPU implementation of spectral-element methods for solving 3D Poisson type equations on rectangular domains and its applications, Communications in Computational Physics, Vol. 36, no. 5 (2024), pp. 1157–1185. DOI PDF arXiv Demonstration for how to run the code
44. S. Zheng, H. Yang, and X. Zhang, On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation. on the special Computational Methods and Models in Deep Learning for Inverse Problems, Journal of Computational and Applied Mathematics 451 (2024), pp. 116053. PDF
43. L. Cross and X. Zhang, On the monotonicity of Q3 spectral element method for Laplacian, Annals of Applied Mathematics 40 (2) (2024), pp. 161–190. PDF doi: 10.4208/aam.OA-2024-0007
42. C. Liu, B. Riviere, J. Shen and X. Zhang, A simple and efficient convex optimization based bound-preserving high order accurate limiter for Cahn--Hilliard--Navier--Stokes system, SIAM Journal on Scientific Computing, 46, no. 3 (2024): A1923-A1948. PDF arXiv
41. M. Dai, M. Hoeller, Q. Peng, and X. Zhang, Kolmogorov's dissipation number and determining wavenumber for dyadic models. Nonlinearity 37, no. 2 (2024): 025015. DOI arXiv
40. Z. Chen, J. Lu, Y. Lu and X. Zhang, On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem. SIAM Journal on Numerical Analysis 62 (2024), pp. 667–691. arXiv
39. L. Cross and X. Zhang, On the monotonicity of Q2 spectral element method for Laplacian on quasi-uniform rectangular meshes, Communications in Computational Physics, Vol. 35, No. 1, pp. 160-180, 2024. PDF doi:10.4208/cicp.OA-2023-0206
38. C. Liu, Y. Gao and X. Zhang, Structure preserving schemes for Fokker-Planck equations of irreversible processes, Journal of Scientific Computing 98(1):4, 2024. arXiv
37. C. Liu and X. Zhang, A positivity-preserving implicit-explicit scheme with high order polynomial basis for compressible Navier–Stokes equations, Journal of Computational Physics 493:112496, 2023. PDF
36. M. Dai, B. Vyas and X. Zhang, 1D Model for the 3D Magnetohydrodynamics. Journal of Nonlinear Science, 33, Article number: 87 (2023). DOI PDF
35. X. Liu, J. Shen and X. Zhang, An efficient and robust SAV based algorithm for discrete gradient systems arising from optimizations, SIAM Journal on Scientific Computing, Vol. 45, No. 5, pp. A2304--A2324, 2023. PDF
34. B. Ren, Z. Gao, Y. Gu, S. Xie, and X. Zhang, A positivity-preserving and well-balanced high order compact finite difference scheme for shallow water equations, Communications in Computational Physics 35 (2024), pp. 524-552. PDF
33. H. Li and X. Zhang, A high order accurate bound-preserving compact finite difference scheme for two-dimensional incompressible flow, Communications on Applied Mathematics and Computation, Volume 6, pages 113–141, (2024). Focused Issue in Memory of Prof. Ching-Shan Chou. PDF https://doi.org/10.1007/s42967-022-00227-9
32. C. Fan, X. Zhang and J. Qiu, Positivity-preserving high order finite difference WENO schemes for the compressible Navier-Stokes equations, Journal of Computational Physics 467 (2022): 111446. PDF

31. J. Hu and X. Zhang, Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations, IMA Journal of Numerical Analysis 43 (2022), pp. 1450–1484. PDF

30. J. Shen and X. Zhang, Discrete Maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation, Communications in Mathematical Sciences, Volume 20 (2022) Number 5, pp.1409-1436. PDF

29. H. Li, D. Appelö and X. Zhang, Accuracy of spectral element method for wave, parabolic and Schrödinger equations, SIAM Journal on Numerical Analysis 60(1):339–363, 2022. PDF See Section 2.8 in Hao Li's thesis for detailed discussion of Neumann b.c..
28. C. Fan, X. Zhang and J. Qiu, Positivity-preserving high order finite volume hybrid Hermite WENO schemes for compressible Navier-Stokes equations, Journal of Computational Physics, Volume 445, 2021, 110596. PDF

27. M. Li, Y. Cheng, J. Shen and X. Zhang, A Bound-Preserving High Order Scheme for Variable Density Incompressible Navier-Stokes Equations, Journal of Computational Physics 425 (2021): 109906. PDF
26. H. Li and X. Zhang, On the monotonicity and discrete maximum principle of the finite difference implementation of C^0-Q^2 finite element method, Numerische Mathematik 145, 437–472 (2020). PDF
25. H. Li and X. Zhang, Superconvergence of high order finite difference schemes based on variational formulation for elliptic equations, Journal of Scientific Computing 82, 36 (2020). PDF See Section 2.8 in Hao Li's thesis for detailed discussion of Neumann b.c..
24. H. Li and X. Zhang, Superconvergence of C^0-Q^k finite element method for elliptic equations with approximated coefficients, Journal of Scientific Computing 82, 1 (2020). PDF
23. H. Li, S. Xie and X. Zhang, A high order accurate bound-preserving compact finite difference scheme for scalar convection diffusion equations, SIAM Journal on Numerical Analysis, 2018, 56(6), 3308-3345. PDF
22. S. Srinivasan, J. Poggie and X. Zhang, A positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations, Journal of Computational Physics, Vol 366, 2018, Pages 120-143. PDF
21. J. Hu, R. Shu and X. Zhang, Asymptotic-preserving and positivity-preserving implicit-explicit schemes for the stiff BGK equation, SIAM Journal on Numerical Analysis, 2018, 56(2), 942–973. PDF
20. J. Hu and X. Zhang, On a class of implicit-explicit Runge Kutta schemes for stiff kinetic equations preserving the Navier-Stokes limit, Journal of Scientific Computing, (2017) 73: 797-818. PDF
19. W. Huang, K. Gallivan and X. Zhang, Solving PhaseLift by low-rank Riemannian optimization methods for complex semidefinite constraints. SIAM Journal on Scientific Computing, 39-5 (2017), pp. B840-B859. CODE. PDF

18. X. Zhang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations, Journal of Computational Physics, 328 (2017): 301–343. DOI. PDF
17. X. Zhang, A curved boundary treatment for discontinuous Galerkin schemes solving time dependent problems, Journal of Computational Physics, 308 (2016): 153-170. DOI. PDF.

16. X. Cai, X. Zhang and J. Qiu, Positivity-preserving high order finite volume HWENO schemes for compressible Euler equations, Journal of Scientific Computing, (2016) 68: 464. DOI. PDF.
15. X. Zhang and S. Tan, A simple and accurate discontinuous Galerkin scheme for modeling scalar-wave propagation in media with curved interfaces, GEOPHYSICS Mar 2015, Vol. 80, No. 2, pp. T83-T89. DOI. PDF.
14. L. Demanet and X. Zhang, Eventual linear convergence of the Douglas Rachford iteration for basis pursuit, Mathematics of Computation 85 (2016), 209-238. DOI. PDF.
13. Y. Xing and X. Zhang, Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes, Journal of Scientific Computing, v57 (2013), pp. 19-41. DOI. PDF.
12. Y. Zhang, X. Zhang and C.-W. Shu, Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes, Journal of Computational Physics, v234 (2013), pp. 295-316. DOI. PDF.
11. X. Zhang, Y.-Y. Liu and C.-W. Shu, Maximum-principle-satisfying high order finite volume WENO schemes for convection-diffusion equations, SIAM Journal on Scientific Computing, v34 (2012), pp.A627-A658. DOI. PDF.
10. X. Zhang and C.-W. Shu, Positivity-preserving high order finite difference WENO schemes for compressible Euler equations, Journal of Computational Physics, v231 (2012), pp.2245-2258. DOI. PDF.
9. X. Zhang and C.-W. Shu, A minimum entropy principle of high order schemes for gas dynamics equations, Numerische Mathematik, (2012) 121:545-563. DOI. PDF.
8. C. Wang, X. Zhang, C.-W. Shu and J. Ning, Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations, Journal of Computational Physics, v231 (2012), pp.653-665. DOI. PDF.
7. X. Zhang and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order schemes for conservation laws: Survey and new developments, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, v467 (2011), pp.2752-2776. DOI. PDF. See Appendix C in this paper for a correction to the proof of Lemma 1.
6. X. Zhang and C.-W. Shu, Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms, Journal of Computational Physics, v230 (2011), pp.1238-1248. DOI. PDF.

5. X. Zhang, Y. Xia and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, Journal of Scientific Computing, v50 (2012), pp.29-62. DOI. PDF.

4. Y. Xing, X. Zhang and C.-W. Shu, Positivity preserving high order well balanced discontinuous Galerkin methods for the shallow water equations, Advances in Water Resources, v33 (2010), pp.1476-1493. DOI. PDF.

3. X. Zhang and C.-W. Shu, On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, Journal of Computational Physics, v229 (2010), pp.8918-8934. DOI. PDF.
2. X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, Journal of Computational Physics, v229 (2010), pp. 3091-3120. DOI. PDF. See Appendix C in this paper for a correction to the proof of Lemma 2.4.

1. X. Zhang and C.-W. Shu, A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws, SIAM Journal on Numerical Analysis, Volume 48, Issue 2 (2010), pp. 772-795. DOI. PDF.

Publications in Refereed Book Chapters and Conference Proceedings

5. X. Zhang, Recent Progress on Qk Spectral Element Method: Accuracy, Monotonicity and Applications, In: Chun, S., Jung, JH., Park, EJ., Shen, J. (eds) Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023. ICOSAHOM 2023. Lecture Notes in Computational Science and Engineering, vol 142. Springer, Cham. (2025), pp 113-137. PDF

4. Z. Xu and X. Zhang, Bound-preserving high order schemes, Volume 18, Handbook of Numerical Methods for Hyperbolic Problems: Applied and Modern Issues, R. Abgrall and C.-W. Shu, Editors, North-Holland, Elsevier, Amsterdam, 2017, pp. 81-102. PDF

3. Wen Huang, Kyle A. Gallivan, Xiangxiong Zhang, Solving PhaseLift by Low-rank Riemannian Optimization Methods, The International Conference on Computational Science 2016, ICCS 2016, 6-8 June 2016, San Diego, California, USA. Procedia Computer Science, Volume 80, 2016, Pages 1125-1134, ISSN 1877-0509, DOI
2. X. Zhang and S. Tan, A simple and accurate discontinuous Galerkin scheme for modeling scalar-wave propagation in media with curved interfaces, in Proc. SEG annual meeting, Denver, October 2014. DOI
1. M. Leinonen, R. J. Hewett, X. Zhang, L. Ying, L. Demanet, High-dimensional wave atoms and compression of seismic datasets, in Proc. SEG annual meeting, Houston, September 2013. DOI. PDF.

Preprints

1. T. Yu, S. Zheng, J. Lu, G. Menon and X. Zhang, Riemannian Langevin Monte Carlo schemes for sampling PSD matrices with fixed rank. PDF arXiv

2. E. Gil Torres, M. Jacobs, and X. Zhang, Asymptotic Linear Convergence of ADMM for Isotropic TV Norm Compressed Sensing, PDF arXiv
3. C. Liu, D. Milesis, C.-W. Shu, and X. Zhang, Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations, PDF arXiv

Ph.D. students theses

2. Shixin Zheng, Riemannian optimization over PSD fixed rank matrix constraints and applications, 2025
1. Hao Li, Accuracy and monotonicity of spectral element method on structured meshes, 2021.

XIANGXIONG ZHANG

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Ph.D. students theses

Books:

Preprints and publications:

Preprints

Publications in Refereed Book Chapters and Conference Proceedings

Publications and Accepted Papers in Refereed Journals:

Publications in Refereed Book Chapters and Conference Proceedings

Preprints

Ph.D. students theses