I-Liang Chern
Gender Male
Education Ph.D., New York University, USA
Tel 886-3-4227151 Ext. 65107
Fax 886-3-4257379
Office Room 415, Hong-Jing Building
Field Scientific ComputingĦBNumerical Partial D ifferential EquationsĦBImage processingĦBPartial Differential Equations
Professor National Chiao Tung University 2012~2014
Director Center of Mathematical Modeling and Scientific Computing
National Chiao Tung University 2012~2014
Professor National Taiwan University 1991-2012
Chairman (Math. Dept) National Taiwan University 1999-2002
Visiting Professor Chinese University of Hong Kong Jan.-June, 2009
Visiting Professor Nat'l Univ. of Singapore summer, 1998
Visiting Professor U.C. Berkeley 1996-1997
Adjunct assistant Prof. U. of Chicago 1990-1991
Assistant Mathematician Argonne National Lab. 1989-1991
Research Assistant Prof. Courant Institute 1987-1989
Research Associate Fellow Academia Sinica 1983-1987
Post-doctoral Fellow MSRI, Berkeley Feb.-Aug. 1984
Distinguished research award, NSC, 1992-1993.
Distinguished Professor, National Taiwan University, 2008-2011.
Distinguished Professor, National Chiao Tung University, 2013-2014


[J1] J-H Chen, I-L. Chern andWW Wang, \A Complete Study of the Ground State Phase Diagrams of Spin-1 Bose-Einstein Condensates in a Magnetic Field via Continuation Methods," Journal of Scienti_c Computing, 2014 (accepted) (SCI IF 1.698).

[J2] Y-C Shu, I-L. Chern* and C-C Chang, \Accurate Gradient Approximation for Complex Interface Problems in 3D by an Improved Coupling Interface Method," Journal of Computational Physics 275 (2014) 642-661. (SCI IF 2.138).

[J3] Liren Lin and I-Liang Chern*, \A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates," Discrete and Continuous Dynamical Systems, Ser. B, 19(4) (2014) 1119-1128. (SCI IF 1.005).

[J4] E. S. Helou, Y. Censor*, T-B Chen, I-L Chern, _AR De Pierro, M Jiang and H H-S Lu, \Stringaveraging expectation maximization for maximum likelihood estimation in emission tomography," Inverse Problems 30 (2014) 055003 (SCI IF 1.896).

[J5] B.-W. Jeng, C.-S. Chien* and I-L. Chern, \Spectral collocation and a two-level continuation scheme for dipolar Bose-Einstein condensates," Journal of Computational Physics 256 (2014) 713-727. (SCI IF 2.138).

[J6] Chern, I-Liang* and Hai-Liang Li, \Long-time behavior of the nonlinear Schrodinger{Langevin equations," Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 8, 505-544 (2013).

[J7] Yi-Cheng Hsu, I-Liang Chern, Wei Zhao, Borjan Gagoski, Thomas Witzel,and Fa-Hsuan Lin*, \Mitigate B11 Inhomogeneity Using Spatially Selective Radio-frequency Excitation with Generalized Spatial Encoding Magnetic Fields," Magnetic Resonance in Medicine 71: 1458-1469 (2014). (SCI IF 3.267).

[J8] Weizhu Bao, I-Liang Chern and Yanzhi Zhang*, "E_cient methods for computing ground states of spin-1 Bose-Einstein condensates based on their characterizations," Journal of Computational Physics 253 (2013) 189-208. (SCI IF 2.310).

[J9] Y. Li, I-L. Chern, J-D. Kim, X. Lin*, \Numerical method of fabric dynamics using front tracking and spring model," Comm. in Comput. Physics, Vol. 14, No. 5 (Nov. 2013) 1228-1251. (SCI IF 2.077)

[J10] P. Chen*, C. Lin and I. Chern, \A Perfect Match Condition for Point-Set Matching Problems Using the Optimal Mass Transport Approach," SIAM J. on Imaging Sciences, 6 (2) (2013) 730-764 (SCI IF 4.656).

[J11] Chun-Hao Teng, I-Liang Chern and Ming-Chih Lai*, \Simulating binary uid-surfactant dynamics by a phase _eld model," Discrete and Continuous Dynamical Systems - Series B , Vol 17, No 4, 1289-1307 (2012). (SCI IF 0.921).

[J12] Yongguei Zhu* and I-Liang Chern, \Convergence of the alternating minimization method for sparse MR image reconstruction," Journal of Information & Computational Science 8:11 (2011) 2067-2075.

[J13] Jen-Hao Chen, I-Liang Chern*,WeichungWang, \Exploring Ground States and Excited States of Spin-1 Bose-Einstein Condensates by Continuation Methods," Journal of Computational Physics, Vol. 230, (2011), 2222-2236.(SCI IF 2.310)

[J14] Daomin Cao, I-Liang Chern, Jun-Cheng Wei*, \On Ground State of Spinor Bose-Einstein Condensates,"NOEDA-Nonlinear Partial Di_erential Equations and Applications Vol. 18, No. 1, (2011), 427-445.(SCI IF 0.538)

[J15] I-Liang Chern and Chun-Hsiung Hsia*, \Dynamic phase transition for Cahn-Hilliard equations in cylindrical geometry," Discrete and Continuous Dynamical System, B, Vol. 16, No. 1 (2011), 173-188.(SCI IF 0.921).

[J16] Yu-Chen Shu, Chiu-Yen Kao, I-Liang Chern*, Chien C. Chang, \Augmented Coupling Interface Method for Solving Eigenvalue Problems with Sign-changed Coe_cients," Journal of Computational Physics, Vol. 229 (2010), 9246-9268.(SCI IF 2.310).

[J17] Chang, Chien-Cheng*, Yu-Chen Shu and I-Liang Chern, \Solving guided wave modes in plasmonic crystals," Phys. Rev. B 78, 035133 (2008). (SCI IF 3.691).

[J18] Chern, I-Liang* and Yu-Chen Shu, \Coupling interface method for elliptic interface problems," Journal of Computational Physics, Vol. 225, No. 2, pp.2138-2174 (2007). (SCI IF 2.310).

[J19] Bao, Weizhu*, I-Liang Chern and Fong Yin Lim, \E_cient and spectrally accurate numerical methods for computing ground and _rst excited states in Bose-Einstein condensates," Journal of Computational Physics, no. 2, pp. 836-854 (2006).(SCI IF 2.310).

[J20] Tzeng, Jengnan, Wen-Liang Huang* and I-Liang Chern, \An asymmetric subspace watermarking method for copyright protection," IEEE Transactions on Signal Processing, Vol. 53, No. 2, pp. 1-9(2005). (SCI IF 2.628).

[J21] I-Liang Chern, Jian-Guo Liu* and Wei-Cheng Wang, \Accurate Evaluation of Electrostatics for Macromolecules in Solution," Methods and Applications of Analysis, Vol. 10, No. 2, pp.309-328 (2003).

[J22] Chang, Qiangshun and I-Liang Chern*, \Acceleration methods for total variation-based image denoising," SIAM J. Sci. Comp., Vol. 25, No. 3, pp. 982-994 (2003).(SCI IF 1.569).

[J23] Zhilin Li, Wei-Cheng Wang, I-Liang Chern and Ming-Chih Lai, \New formulation for interface problems in polar coordinates," SIAM J. Sci. Comp., Vol. 25, No. 1, pp. 224-245 (2003).(SCIIF 1.569)

[J24] Tzeng, Jengnan, Wen-Liang Huang* and I-Liang Chern, \Enhancing image watermarking methods with/without reference images by optimization second-order statistics," IEEE Transactions on Image Processing, Vol. 11, No. 7, pp. 771-782(2002). (SCI IF 3.042)

[J25] Chern, I-L.* and C.-C. Yen, \Di_erence wavelet { theory and a comparison study," Methods and Applications of Analysis, Vol. 9, No. 4, pp. 469-492 (2002).

[J26] Chern, I-L.*, \Local and global interaction for nongenuinely nonlinear hyperbolic systems of conservation laws," Indiana University Mathematics Journal, 49 No. 3 (2000) 1199-1228. (SCI IF 1.10)

[J27] Chern, I-L.* and Ming Mei, \Asymptotic stability of critical viscous shock wave for a degenerate hyperbolic conservation law," Comm. Partial Di_erential Equations 23 (1998) 869-886 (SCI IF 0.894)

[J28] Chern, I-L.*, \Long-time e_ect of relaxation for hyperbolic conservation laws," Comm. Math. Phys. 172 (1995) 39-55. (SCI IF 1.941)

[J29] Yang, X.-L., I-L. Chern, N. Zabusky, R. Samtaney and J. Hawley, \Vorticity generation and evolution in shock-accelerated density-strati_ed interfaces," Phys. Fluid A 4, no. 7 (1992) 1531- 1540. (SCI IF 1.926)

[J30] Chern, I-L. and I. Foster, \Parallel implementation of a control method for solving PDEs on the sphere," Parallel Processing for Scienti_c Computing, 301-306, SIAM, Philadelphia, PA, (1992).

[J31] Chern, I-L., T. Colin and H. Kaper, \Classical solutions of the nondivergent barotropic equations on the sphere," Comm. in Partial Di_erential Equations 17 no.5 & 6, (1992) 1001-1019. (SCI IF 0.894)

[J32] Chern, I-L., \Multiple-mode di_usion waves for viscous nonstrictly hyperbolic conservation laws," Comm. Math. Phys. 138 (1991) 51-61. (SCI IF 1.941)

[J33] Chern, I-L., \Large-time behavior of solutions of Lax-Friedrichs _nite di_erence equations for hyperbolic systems of conservation laws," Math. Comp. 56, no. 193 (1991) 107-118. (SCI IF 1.313)

[J34] Yang, X.-L., N. Zabusky, and I-L. Chern, \Breakthrough via Vortex dipoles in shock accelerated density strati_ed layers," Phys. Fluid A 2 no.6 (1990), 892-895. (SCI, 1.926)

[J35] Chern, I-L., \Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for ows with strong discontinuities," Comm. Pure Appl. Math. 42 (1989), 815-844. (SCI IF 2.575).

[J36] Chern, I-L. and T.-P. Liu, \Erratum, Convergence to di_usion waves of solutions for viscous conservation laws," Comm. Math. Phys. 120 (1989), 525-527. (SCI IF 1.941).

[J37] Chern, I-L. and T.-P. Liu, \Convergence to di_usion waves of solutions for viscous conservation laws," Comm. Math. Phys. 110 (1987), 503-517. (SCI IF 1.941).

[J38] Chern, I-L. and P. Colella, A conservative front tracking method for hyperbloic conservation laws, UCRL-97200, Lawrence Livermore National Laboratory, Livermore, CA, (1986).

[J39] Chern, I-L., J. Glimm, O. McBryan, B. Plohr, and S. Yaniv, \Front tracking for gas dynamics," J. Comp. Phys. 62 (1986) 83-110. (SCI IF 2.310).

[J40] W. Miranker and I-L. Chern, \Dichotomy and conjugate gradients in the sti_ initial value problem," Lin. Alg. Appl. 36 (1981)(SCI IF 0.974).



[B1] YL Zhu, XN Wu, IL Chern and ZZ Sun, Derivative Securities and Di_erence Methods, 2nd edition, Springer Finance (2013) pp. 1-647.

[B2] You-lan Zhu, Xiaonan Wu and I-Liang Chern, \Derivative Securities and Di_erence Methods," Springer, (2004) pp. 1-513. (google citation 36)



[R1] Liren Lin and I-Liang Chern, \On a phase transition phenomenon of the ground states of spin-1 Bose-Einstein condensates," ArXiv:1302.0279, 2013.



[L1] I-Liang Chern, Financial Mathematics, pp.1-108, (1998).

[L2] I-Liang Chern, Mathematical Modeling and Ordinary Di_erential Equations, pp. 1-155, (2002).

[L3] I-Liang Chern, Finite Di_erence Methods for Partial Di_erential Equations, pp. 1-121 (2004).

[L4] I-Liang Chern and Jun Zou, Lecture Notes on Computational and Applied Mathematics, pp. 1-158 (2010).

[L5] I-Liang Chern, Methods in Applied Mathematics pp. 1-71 (2012).

[L6] I-Liang Chern, Applied Analysis, pp. 1-176 (2013).


Selected Invited Talks: 2009-2014

1. Ground State Patterns and Phase Transitions for Spin-1 Bose-Einstein Condensates, Annual Meeting of Japan Society for Industrial and Applied Mathematics, 3 September 2014, Tokyo, Japan.

2. Ground States of Spin-1 Bose-Einstein Condensates w/o external magnetic _eld, International Conference on the Mathematical Theory of Liquid Crystals and Related Topics, 19 June 2014, NYU-Shanghai, China. Bose-Einstein Condensates w/o external magnetic _eld, 2014 Japan- Taiwan Joint Workshop on Numerical Analysis and Scienti_c Computation6 April 2014, Kyoto University, Japan.

3. On Ground States of Spin-1 Bose-Einstein Condensates w/o external magnetic _eld, IMS Workshop on Nonlinear PDEs from Fluids and Related Topics, 24-26 March 2014, Chinese University of Hong Kong, Hong Kong.

4. On Ground States of Spin-1 Bose-Einstein Condensates w/o external magnetic _eld, 2013 NCTS Mathematics Physics Joint Colloquium, Sep. 26, 2013, Taipei.

5. On Ground States of Spin-1 Bose-Einstein Condensates with/without external magnetic _eld, International Congress of Chinese Mathematicians, 14-19, 2013, Taipei.

6. Exploring Ground States and Excited States of Spin-1 Bose-Einstein Condensates with/without External Magnetic Field, Workshop on \Con_ned Quantum Systems: Modeling, Analysis and Computation", Feb. 4 - 8, 2013, Wolfgang Pauli Institut, Vienna, Austria.

7. Coupling Interface Method and Macromolecules in Ionic Solution, A workshop in honor of Stanley Osher, December 15-18, 2012, Tsing Hua, Beijing, China.

8. Exploring ground states and excited states of Spin-1 Bose-Einstein condensates, International Conference on Mathematical Modeling, Analysis and Computation, Xiamen, June 22-25, 2012.

9. Accurate Gradient Approximation at Interfaces by Coupling Interface Method for Elliptic Interface Problems, 2011 International Conference on Applied Mathematics and Interdisciplinary Research, Chern Institute of Mathematics, Nankai Univ., June 13-16, 2011.

10. Characterization of ground states of spin-1 Bose-Einstein condensates, The seventh International

Congress on Industrial and Applied Mathematics, Vancouver, July 18-22, 2011.

11. Exploring ground states and excited states of Spin-1 Bose-Einstein condensates by Continuation method, International Conference on Applied Mathematics, Hong Kong, June 7-11, 2010

12. Parallel MR imaging, SIAM Conference on Image Science, Chicago, Apr. 12-14, 2010

13. Two Finite Di_erence Methods for Solving Poisson-Boltzmann Equation, International Workshop

on Continuum Modeling of Biomolecules, Beijing, September 14-16, 2009.


Research Summary

I-Liang Chern, 2014/09/15

Publications: 44, citations: 1075, i10-index: 18, H-index: 15 (cited from google scholar search September, 2014,




Bose-Einstein Condensates and Nonlinear Schrodinger equations: In the past _ve years, my major research topic is on exploring the ground states of spinor Bose-Einstein condensations (BEC), both numerically and analytically. I chose this topic because BEC was only realized experimentally in 1995 and earned a Nobel prize in 2000. This means that it is important from physicists' point of view and its mathematical theory is still quite unexplored. The spinor BECs involve vector wave function and nonlinear Schrodingier systems. In many cases, the corresponding energy functional and the constraints may not be convex. Thus, its mathematical theory is more challenging and less explored, while the physical phenomena are rich. Furthermore, the mathematical theory for Schrodingier systems is closely related to nonlinear optics and superconductivities, which are also very important in application. Thus, the mathematical theory of BECs is fundamental. 

Joint with J-H Chen and W Wang, we developed a pseudo-arc length continuation method to compute the ground states and excited states for spinor BECs. From computational results, we are able to

characterize the ground state patterns and also observed component separation for excited states in ferro-magnetic systems. This work was published on J. Comp. Phys.[13]. More recently, we continue this work and provide a complete numerical study of the ground state patterns and phase transitions for spin-1 BECs in uniform magnetic _eld. This work was just accepted by J. Scienti_c Computing. My contribution was to propose the problem, help to develop numerical method, and to interpret computational results. 

Inspired by our computational simulations, I began analytic studies of the ground state patterns and their phase transitions. In a joint work with Cao and Wei, we obtained existence and nonexistence theorems for ground states of spin-1 BECs in one space dimension. Later, jointly with my Ph.D student, Liren Lin, we showed that the ground state has to be so-called single mode approximation (SMA) in physics literature for ferromagnetic systems, whereas it becomes a two-component BEC for antiferromagnetic systems. The key step is a kinetic-energy reduced mass-redistribution lemma. I expect it will be important in the community of mathematical physics. This paper was published on the Discrete and Continuous Dynamical Systems, Ser. B (2014). My contribution was to propose the problem and to point out the importance of a general form of this mass-redistribution lemma. A directly application of this ground state characterization is to develop a fast algorithm for computing ground states because we can minimize the energy functional over a smaller class. This was a joint work with W-Z Bao and Y-Z Zhang, in which we study the ground states of spin-1 BECs in the Io_e-Pritchard magnetic _eld. In this study, we also discovered that the ground state has to be SMA as long as the I-P _eld exists. This work was published on J. Comput. Phys 2013. Later, jointly with Liren Lin again, we applied this lemma and discovered more properties of ground states for antiferromagnetic spinor BECs in uniform magnetic _eld in three dimensions. These include a strict pointwise monotone property between di_erent hyper_ne states, a strict monotonic property of ground state energy functional as a function of total magnetization and applied magnetic _eld strength. In addition, we proved a phase transition from 2C state to 3C state as the applied _eld strength increased. This phenomenon was observed both experimentally and numerically (also in our computational works). The proof is no easy at all. This was part of Liren Lin's Ph.D thesis. This work was available on ArXiv.

There are two side works related to my research on BECs. One is a joint work with B-W Jeng and C-S Chien, where we propose a spectral collation method to study the ground states of dipolar Bose-Einstein condensates. In this case, a long-range dipole-dipole interaction is important and is handled by introducing a Poisson equation. The ground states and vortex states are found through a two-level continuation method. Extensive numerical experiments in 3D are reported. My role was to propose the problem and to interpret the numerical results. This work was published on J. Comput. Phys.

The other side work was jointly with Hailiang Li on the asymptotic behaviors of the solutions of the Schrodinger-Langevin equation. It is shown that the momentum damping overwhelms the quantum dispersion and the solution tends pointwisely to a nonlinear di_usion wave. The new analytic trick, which is di_erent from my earlier work on di_usion wave, is an energy estimate associated with the quantum dispersion. Both Hailiang Li and I have equal contribution.

Elliptic Interface Problems and Phase Field Models for Multi-phase Media: This is a continuation of my earlier work on this subject. Two papers were published in this period. One is a joint work with the front tracking team from Stony Brook. We have merged our Coupling Interface Method (CIM) 3D code with their FronTier Code and have studied fabric dynamics [9].

The second work is an improvement of the CIM. Our original CIM was proposed in 2007 and has attracted much attention since then (47 citations so far). In our earlier CIM, the second order method cannot be applied at some exceptional grid points when the underlying interface is very complex, and a hybrid method was proposed. Although the overall error is still second order through least squares _tting, the absolute errors uctuate at di_erent mesh sizes. In our new work, we proposed two recipes to avoid these annoying errors. It is second order accurate everywhere, and it can handle quite complex interfaces in three dimensions without error uctuation. We got an impressive comment from referees. This paper was just accepted by J. Comput. Physics. My contribution was one of the recipes.

The elliptic interface problem is a research topic of multi-phase physics. For the latter, I also study the phase _eld model. Jointly with CH Hsia, we study dynamic phase transition of binarysystem in cylindrical geometry. It is found that the phase separation in pan-cake geometry forms a pattern with four components, instead of two. This analytic result surprises many peoples[15]. Jointly with CH Teng and MC Lai [11], we propose a phase _eld model for binary uids with surfactant. We characterize analytically the structure of interfaces, comparing with physical experiment and performing numerical simulations by spectral method. It is shown that the surfactant favors the creation of interfaces and stabilizes the formation of phase regions (cited 26).

Compressed Sensing and Image processing The compressive sensing is very hot in the _elds of applied mathematics, statistical science, computer science, electric engineering, etc., yet there is almost no people studying this subject in the community of applied mathematics. So, I started to look into this _eld since 2008, and tried to attract some young people and students entering this _eld. Joint with some young people, we published few papers. In a joint work with Pengwen Chen and Ching-Long Lin [10], we provide a registration method for solving point set matching problems. The method is a combination of a global a_ne transform and a local curl-free transform. The curl-free transform is estimated by optimizing some kernel correlation function weighted by a square root of a pair of correspondence matrices, which can be regarded as an approximation of the mass transport problem. We apply this method to match two sets of lung branch points whose displacement is caused by lung volume changes. Nearly perfect match performances verdict the e_ectiveness of this model. This paper was published on SIAM J. on Image Sciences.

In a joint project with engineers, my student studies how to mitigate B+ 1 inhomogeneity in a high-_eld magnetic resonance imaging (MRI). Such inhomogeneity causes spatially dependent contrast and makes clinical diagnosis di_cult. The proposed method is a two-step design procedure in which (a) a combination of linear and quadratic spatial encoding magnetic _eld is used to remap the B+ 1 map in order to reduce the inhomogeneity problem to one dimension, (b) the locations, the amplitudes and the phases of spokes are estimated in one dimension. It is shown both numerically and experimentally that this design can mitigate the B+ 1 inhomogeneity at 7T e_ciently. This work was published on Magnetic Resonance in Medicine (2013).

In another work with Censor, TB Chen, De Pierro, Helou, Jiang and Lu [4], we study the Expectation-Maximization (EM) algorithm for Maximum Likelihood Estimation (MLE) in Positron Emission Tomography (PET) and propose a new algorithmic structure called the String-Averaging Expectation-Maximization (SA-EM). In our simulation study, high-contrast and less noisy images with clear object boundaries are reconstructed with the proposed SA-EM algorithm in less computation time. Together with the new scheme, we propose a stopping criterion for this and other fast algorithms in tomography based on the curvature of the likelihood, as well as an L-curve to analyze iterations quality. Also, we present new convergence results for this family of algorithms. 

In a joint work with Yungguei Zhu [12], we applied the alternating minimization method with total variation regularization and wavelet sparsity for sparse magnetic resonance image reconstruction. Numerical experiments were performed and convergence theorem is proved. The results show that radial compressed sensing is feasible for MR imaging.





Elliptic Interface Problems: One of my research direction in this period is numerical studies of elliptic interface problems. This was motivated by a joint work with Jian-Guo Liu and Wei- Cheng Wang about 10 years ago on computing electrostatic _eld for macromolecules in ionic solvent, an important problem arisen from drug design. In that work, we realized that a high order solver for elliptic interface problem is needed. Jointly with my Ph.D student Yu-Chen Shu, we proposed the coupling interface method (CIM) to solve this elliptic interface problems numerically[18]. Our method is for arbitrary dimensions. It is second order accurate not only for the unknown u, but also for its gradient. It is also capable to handle high contrast and complex interface problems which are known to be very di_cult numerically. Our numerical results show that this method is superior to many other computational interface methods. The work was published on the J. Comput. Phys. and has attracted many citations so far (47 citations). Some group in Europe has applied it to solve parabolic interface problems. 

We have also applied this method to solve surface plasma problems [17, 16]. Surface plasmonic waves are EM waves propagating near the metal-dielectric interfaces. They are important for biosensor design, superlens study, etc. in optical science and engineering. It is di_cult to compute these waves due to their con_ned and oscillatory behaviors near the interface. Using CIM, we were able to compute them and analyze their optical properties. The works were published on J. Comput. Phys. and Phys. Rev. B.




Wavelets and image processing: I propose \di_erence wavelet"[25] which is based on _nite di_erence. The corresponding analytic _lter is the shortest, whereas the synthesis part can still be fast through a cyclic reduction method. A thorough mathematical theory was also developed. In a joint work with Tseng and Huang, a digital water marking method was proposed where the water mark is added in an unimportant direction of the original image. The unimportant direction is determined by singular value decomposition method. The proposed method is robust and hard to be broken. The works were published on IEEE Journals[24, 20] (cited 48) For image denoising, jointly with Qianshun Chang, we proposed three acceleration methods based on the total variation penalty approach. We get linear complexity. It is superior to many existing methods. The work was published on SIAM J. Sci. Comput[22] (cited 49).

Interface problems: A fast solver for Poisson-Boltzmann equation with discontinuous coe_- cients was proposed. The method is second-order and of linear complexity, whereas most existing method is _rst-order. The work was published on Methods and Applications of Analysis[21] (cited 40).

Finite Di_erence Methods in Financial Mathematics: I write a book with Zhu and Wu on \Derivative Security and Di_erence Methods." This is a 513 pages book published by Springer-Verlog[2] (cited 54).


Before 2000


Computational Fluid Dynamics

- Front tracking methods: Jointly with Glimm et al. [39](cited 275), we have developed the _rst successful, general-purpose front tracking code for solving gas dynamic problems such as shock di_raction problem, Rayleigh-Taylor problem, Richtmyer-Meshkov problem, etc. Jointly with P. Colella [38] (cited 112 times), we proposed the _rst \conservative" front tracking method for gas dynamics in two dimension. This method has been used by many researchers (e.g., Colella, Majda, Berger, etc.) to various applications (e.g., combustion ow calculations).

- Geouid dynamics on the sphere. I proposed two versions of control-volume method on icosahedral grid for the shallow-water equations on the sphere. Its parallel implementation was also done together with I. Foster[30]. The purpose is to develop advanced fast and high resolution general circulation model for climate study. This method can achieve almost linear speedup.

- Non-divergent barotropic equations on the sphere The non-divergent barotropic equations on a sphere provide the simplest, yet the most important mathematical model for the description of large-scale horizontal motions of the atmosphere. Based on Holder estimation, we prove global existence, uniqueness, and regularity theorems for the non-divergent barotropic equations. This paper was published on Comm in P.D.E.[31].

- Mixing of Fluid Flows: This was a joint work with N. Zabusky. We studied the mixing process in a density-strati_ed uid induced by a strike of a shock. High order Godunov method was used. A convective \breakthrough" phenomenon was observed and quanti_ed. The phenomenon was interpreted as a dipolar-vortex dynamics. This work also gave an insight of the well-known Richtmyer-Meshkov instability in the mixing uid ow, namely, the evolution of a shock-accelerated density-strati_ed interface was determined by (1) the vorticity deposition by the shock-interface interaction (zeroth order e_ect) and(2) the interaction of the interface with the vortex sheets shed from the transmitted and reected waves (_rst order e_ect) [34, 29].


 Hyperbolic Conservation Laws:

- Global existence for large initial data: I proved a global existence of ow that is a perturbation of a strong discontinuity [35]. This was a generalization of Glimm's famous work in 1965, where the ow is a perturbation of a constant state. This work answered part of the well-known problem involving the global existence of a solution for hyperbolic conservation laws with large data. In another work, I proved the global existence for non-genuinely nonlinear hyperbolic systems [26]. This work took me about 10 years to _nish it. Their citations are 41 and 10 respectively.

- Viscous conservation laws

In a joint work with TP Liu [37], we identi_ed that the time-asymptotic solutions of viscous conservation laws were the di_usion waves. These waves are important because they carry the invariant masses. The optimal convergent rate to these waves was also identi_ed. This work was for the case that the inviscid part of the equations are strictly hyperbolic. A generalization to the nonstrictly hyperbolic systems was done by myself [32]. Nonstrictly hyperbolic systems occur in some phase transition models. In this case, \multiple-mode di_usion waves" were the time-asymptotic solutions. Both papers were published on Comm. in Math Phys with citation 59 and 28 times. 

- Hyperbolic conservation laws with relaxation

The hyperbolic conservation laws with relaxation appear in many physical systems such as non-equilibrium gas dynamics, ood ow with friction, visco-elasticility, magnetohydrodynamics, etc. I have shown that the long-range e_ect of relaxation is equivalent to a viscous e_ect. As a consequence, the long-time behavior of solutions of such systems consist of di_usion waves. The convergent rate to these di_usion waves is also obtained [28]. This work has been cited 62 times.


Numerical Partial Di_erential Equations

- I proved a stability theorem and analyzed truncation errors for a front tracking method for ows with strong discontinuities in one space dimension [35]. This result is the only analytical result about the front tracking method. It gives insight of wave-interaction picture near fronts.(cited 33) 

- I identi_ed that the major error source of the Lax-Friedrichs scheme was the discrete di_usion waves. They were also the time-asymptotic of the Lax-Friedrichs _nite di_erence equations. Construction and properties of these discrete di_usion were also given [33] (cited 9). 


Chao A. Hsiung
Yuan-Shih Chow
Ju-Kwei Wang
Men-Chang Hu
Tsong-Chih Shieh
Yang Hua
Ching-Song Chou
I-Shou Chang
Hung-Yih Chen
Hwei-Mei Ko
Chang-Shou Lin
Chin-Lung Wang
Hui-Wen Lin
Hua-Min Huang
Liang-Chung Hsia
Po-Hung Liu
Chiang Lin
I-Liang Chern
Shing-Whu Jha
Hong-Chang Lee
I-Feng Chao
Chih-Hung Chang
Chien-Chang Yen
Copyright © 2006 NCU Department of Mathematics All Rights Reserved.