Selected Publications 
JOURNAL PUBLICATIONS
[J1] JH Chen, IL.
Chern andWW Wang, \A Complete Study of the Ground State Phase Diagrams of Spin1
BoseEinstein Condensates in a Magnetic Field via Continuation Methods," Journal
of Scienti_c Computing, 2014 (accepted) (SCI
IF
1.698).
[J2] YC Shu, IL.
Chern* and CC Chang, \Accurate Gradient Approximation for Complex Interface
Problems in 3D by an Improved Coupling Interface Method," Journal of
Computational Physics
275
(2014) 642661. (SCI
IF
2.138).
[J3] Liren Lin and
ILiang Chern*, \A kinetic energy reduction technique and characterizations of
the ground states of spin1 BoseEinstein condensates," Discrete and Continuous
Dynamical Systems, Ser. B,
19(4)
(2014) 11191128. (SCI
IF
1.005).
[J4] E. S. Helou,
Y. Censor*, TB Chen, IL Chern, _AR De Pierro, M Jiang and H HS 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 IL. Chern, \Spectral collocation and a twolevel continuation
scheme for dipolar BoseEinstein condensates," Journal of Computational Physics
256
(2014) 713727. (SCI
IF
2.138).
[J6] Chern,
ILiang* and HaiLiang Li, \Longtime behavior of the nonlinear
Schrodinger{Langevin equations," Bulletin of the Institute of Mathematics,
Academia Sinica, Vol. 8, 505544 (2013).
[J7] YiCheng Hsu,
ILiang Chern, Wei Zhao, Borjan Gagoski, Thomas Witzel,and FaHsuan Lin*,
\Mitigate B11 Inhomogeneity Using Spatially Selective Radiofrequency Excitation
with Generalized Spatial Encoding Magnetic Fields," Magnetic Resonance in
Medicine
71:
14581469 (2014). (SCI
IF
3.267).
[J8] Weizhu Bao,
ILiang Chern and Yanzhi Zhang*, "E_cient methods for computing ground states of
spin1 BoseEinstein condensates based on their characterizations," Journal of
Computational Physics
253
(2013) 189208. (SCI
IF
2.310).
[J9] Y. Li, IL.
Chern, JD. Kim, X. Lin*, \Numerical method of fabric dynamics using front
tracking and spring model," Comm. in Comput. Physics, Vol. 14, No. 5 (Nov. 2013)
12281251. (SCI
IF
2.077)
[J10] P. Chen*, C.
Lin and I. Chern, \A Perfect Match Condition for PointSet Matching Problems
Using the Optimal Mass Transport Approach," SIAM J. on Imaging Sciences, 6 (2)
(2013) 730764 (SCI
IF
4.656).
[J11] ChunHao Teng,
ILiang Chern and MingChih Lai*, \Simulating binary uidsurfactant dynamics by
a phase _eld model," Discrete and Continuous Dynamical Systems  Series B , Vol
17, No 4, 12891307 (2012). (SCI
IF
0.921).
[J12] Yongguei Zhu*
and ILiang Chern, \Convergence of the alternating minimization method for
sparse MR image reconstruction," Journal of Information & Computational Science
8:11 (2011) 20672075.
[J13] JenHao Chen,
ILiang Chern*,WeichungWang, \Exploring Ground States and Excited States of
Spin1 BoseEinstein Condensates by Continuation Methods," Journal of
Computational Physics, Vol. 230, (2011), 22222236.(SCI
IF
2.310)
[J14] Daomin Cao,
ILiang Chern, JunCheng Wei*, \On Ground State of Spinor BoseEinstein
Condensates,"NOEDANonlinear Partial Di_erential Equations and Applications Vol.
18, No. 1, (2011), 427445.(SCI
IF
0.538)
[J15] ILiang Chern
and ChunHsiung Hsia*, \Dynamic phase transition for CahnHilliard equations in
cylindrical geometry," Discrete and Continuous Dynamical System, B, Vol. 16, No.
1 (2011), 173188.(SCI
IF
0.921).
[J16] YuChen Shu,
ChiuYen Kao, ILiang Chern*, Chien C. Chang, \Augmented Coupling Interface
Method for Solving Eigenvalue Problems with Signchanged Coe_cients," Journal of
Computational Physics, Vol. 229 (2010), 92469268.(SCI
IF
2.310).
[J17] Chang, ChienCheng*,
YuChen Shu and ILiang Chern, \Solving guided wave modes in plasmonic
crystals," Phys. Rev. B
78,
035133 (2008). (SCI
IF
3.691).
[J18] Chern,
ILiang* and YuChen Shu, \Coupling interface method for elliptic interface
problems," Journal of Computational Physics, Vol. 225, No. 2, pp.21382174
(2007). (SCI
IF
2.310).
[J19] Bao, Weizhu*,
ILiang Chern and Fong Yin Lim, \E_cient and spectrally accurate numerical
methods for computing ground and _rst excited states in BoseEinstein
condensates," Journal of Computational Physics, no. 2, pp. 836854 (2006).(SCI
IF
2.310).
[J20] Tzeng,
Jengnan, WenLiang Huang* and ILiang Chern, \An asymmetric subspace
watermarking method for copyright protection," IEEE Transactions on Signal
Processing, Vol. 53, No. 2, pp. 19(2005). (SCI
IF
2.628).
[J21] ILiang Chern,
JianGuo Liu* and WeiCheng Wang, \Accurate Evaluation of Electrostatics for
Macromolecules in Solution," Methods and Applications of Analysis, Vol. 10, No.
2, pp.309328 (2003).
[J22] Chang,
Qiangshun and ILiang Chern*, \Acceleration methods for total variationbased
image denoising," SIAM J. Sci. Comp., Vol. 25, No. 3, pp. 982994 (2003).(SCI
IF
1.569).
[J23] Zhilin Li,
WeiCheng Wang, ILiang Chern and MingChih Lai, \New formulation for interface
problems in polar coordinates," SIAM J. Sci. Comp., Vol. 25, No. 1, pp. 224245
(2003).(SCIIF
1.569)
[J24] Tzeng,
Jengnan, WenLiang Huang* and ILiang Chern, \Enhancing image watermarking
methods with/without reference images by optimization secondorder statistics,"
IEEE Transactions on Image Processing, Vol. 11, No. 7, pp. 771782(2002). (SCI
IF
3.042)
[J25] Chern, IL.*
and C.C. Yen, \Di_erence wavelet { theory and a comparison study," Methods and
Applications of Analysis, Vol. 9, No. 4, pp. 469492 (2002).
[J26] Chern, IL.*,
\Local and global interaction for nongenuinely nonlinear hyperbolic systems of
conservation laws," Indiana University Mathematics Journal,
49
No. 3 (2000)
11991228. (SCI
IF
1.10)
[J27] Chern, IL.*
and Ming Mei, \Asymptotic stability of critical viscous shock wave for a
degenerate hyperbolic conservation law," Comm. Partial Di_erential Equations
23
(1998) 869886 (SCI
IF
0.894)
[J28] Chern, IL.*,
\Longtime e_ect of relaxation for hyperbolic conservation laws," Comm. Math.
Phys.
172
(1995) 3955. (SCI
IF
1.941)
[J29] Yang, X.L.,
IL. Chern, N. Zabusky, R. Samtaney and J. Hawley, \Vorticity generation and
evolution in shockaccelerated densitystrati_ed interfaces,"
Phys. Fluid A
4,
no. 7 (1992) 1531 1540. (SCI
IF
1.926)
[J30] Chern, IL.
and I. Foster, \Parallel implementation of a control method for solving PDEs on
the sphere," Parallel Processing for Scienti_c Computing, 301306, SIAM,
Philadelphia, PA, (1992).
[J31] Chern, IL.,
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)
10011019. (SCI
IF
0.894)
[J32] Chern, IL.,
\Multiplemode di_usion waves for viscous nonstrictly hyperbolic conservation
laws,"
Comm. Math. Phys.
138
(1991) 5161. (SCI
IF
1.941)
[J33] Chern, IL.,
\Largetime behavior of solutions of LaxFriedrichs _nite di_erence equations
for hyperbolic systems of conservation laws,"
Math. Comp.
56,
no. 193 (1991) 107118. (SCI
IF
1.313)
[J34] Yang, X.L.,
N. Zabusky, and IL. Chern, \Breakthrough via Vortex dipoles in shock
accelerated density strati_ed layers,"
Phys. Fluid A
2
no.6
(1990), 892895. (SCI,
1.926)
[J35] Chern, IL.,
\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), 815844. (SCI
IF
2.575).
[J36] Chern, IL.
and T.P. Liu, \Erratum, Convergence to di_usion waves of solutions for viscous
conservation laws,"
Comm. Math. Phys.
120
(1989), 525527. (SCI
IF
1.941).
[J37] Chern, IL.
and T.P. Liu, \Convergence to di_usion waves of solutions for viscous
conservation laws,"
Comm. Math. Phys.
110
(1987), 503517. (SCI
IF
1.941).
[J38] Chern, IL.
and P. Colella, A conservative front tracking method for hyperbloic conservation
laws, UCRL97200, Lawrence Livermore National Laboratory, Livermore, CA, (1986).
[J39] Chern, IL.,
J. Glimm, O. McBryan, B. Plohr, and S. Yaniv, \Front tracking for gas dynamics,"
J.
Comp. Phys.
62
(1986) 83110. (SCI
IF
2.310).
[J40] W. Miranker
and IL. Chern, \Dichotomy and conjugate gradients in the sti_ initial value
problem,"
Lin. Alg. Appl.
36
(1981)(SCI
IF
0.974).
BOOKS
[B1] YL Zhu, XN Wu,
IL Chern and ZZ Sun, Derivative Securities and Di_erence Methods, 2^{nd}
edition, Springer Finance (2013) pp. 1647.
[B2] Youlan Zhu,
Xiaonan Wu and ILiang Chern, \Derivative Securities and Di_erence Methods,"
Springer, (2004) pp. 1513. (google citation 36)
REPORTS
[R1] Liren Lin and
ILiang Chern, \On a phase transition phenomenon of the ground states of spin1
BoseEinstein condensates," ArXiv:1302.0279, 2013.
LECTURE NOTES
[L1] ILiang Chern,
Financial Mathematics,
pp.1108, (1998).
[L2] ILiang Chern,
Mathematical Modeling and Ordinary Di_erential Equations,
pp. 1155, (2002).
[L3] ILiang Chern,
Finite Di_erence Methods for Partial Di_erential Equations,
pp. 1121 (2004).
[L4] ILiang Chern
and Jun Zou,
Lecture Notes on Computational and Applied Mathematics,
pp. 1158 (2010).
[L5] ILiang Chern,
Methods in Applied Mathematics
pp. 171 (2012).
[L6] ILiang Chern,
Applied Analysis,
pp. 1176 (2013).
Selected
Invited Talks: 20092014
1. Ground State
Patterns and Phase Transitions for Spin1 BoseEinstein Condensates, Annual
Meeting of Japan Society for Industrial and Applied Mathematics, 3 September
2014, Tokyo, Japan.
2. Ground States of
Spin1 BoseEinstein Condensates w/o external magnetic _eld, International
Conference on the Mathematical Theory of Liquid Crystals and Related Topics, 19
June 2014, NYUShanghai, China. BoseEinstein 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 Spin1 BoseEinstein Condensates w/o external magnetic _eld, IMS Workshop on
Nonlinear PDEs from Fluids and Related Topics, 2426 March 2014, Chinese
University of Hong Kong, Hong Kong.
4. On Ground States
of Spin1 BoseEinstein Condensates w/o external magnetic _eld, 2013 NCTS
Mathematics Physics Joint Colloquium, Sep. 26, 2013, Taipei.
5. On Ground States
of Spin1 BoseEinstein Condensates with/without external magnetic _eld,
International Congress of Chinese Mathematicians, 1419, 2013, Taipei.
6. Exploring Ground
States and Excited States of Spin1 BoseEinstein 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 1518, 2012, Tsing Hua, Beijing, China.
8. Exploring ground
states and excited states of Spin1 BoseEinstein condensates, International
Conference on Mathematical Modeling, Analysis and Computation, Xiamen, June
2225, 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
1316, 2011.
10.
Characterization of ground states of spin1 BoseEinstein condensates, The
seventh International
Congress on
Industrial and Applied Mathematics, Vancouver, July 1822, 2011.
11. Exploring
ground states and excited states of Spin1 BoseEinstein condensates by
Continuation method, International Conference on Applied Mathematics, Hong Kong,
June 711, 2010
12. Parallel MR
imaging, SIAM Conference on Image Science, Chicago, Apr. 1214, 2010
13. Two Finite
Di_erence Methods for Solving PoissonBoltzmann Equation, International Workshop
on Continuum
Modeling of Biomolecules, Beijing, September 1416, 2009.
Research
Summary
ILiang Chern,
2014/09/15
Publications: 44,
citations: 1075, i10index: 18, Hindex: 15
(cited from google
scholar search September, 2014,
http://scholar.google.com/citations?hl=en&user=z2sUUAAAAJ).
20092014
BoseEinstein
Condensates and Nonlinear Schrodinger equations:
In the past _ve years, my major research topic is on exploring the ground states
of spinor BoseEinstein 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 JH Chen
and W Wang, we developed a pseudoarc 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 ferromagnetic 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 spin1 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 spin1 BECs in
one space dimension. Later, jointly with my Ph.D student, Liren Lin, we showed
that the ground state has to be socalled single mode approximation (SMA) in
physics literature for ferromagnetic systems, whereas it becomes a twocomponent
BEC for antiferromagnetic systems. The key step is a kineticenergy
reduced massredistribution 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
massredistribution 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 WZ Bao and YZ Zhang, in which we study the ground states
of spin1 BECs in the Io_ePritchard magnetic _eld. In this study, we also
discovered that the ground state has to be SMA as long as the IP _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 BW Jeng and CS
Chien, where we propose a spectral collation method to study the ground
states of dipolar BoseEinstein condensates. In this case, a longrange
dipoledipole interaction is important and is handled by introducing a Poisson
equation. The ground states and vortex states are found through a twolevel
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
SchrodingerLangevin 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 Multiphase 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 multiphase 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 pancake 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 ChingLong 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 curlfree transform. The
curlfree 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 twostep 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 ExpectationMaximization (EM) algorithm for Maximum Likelihood Estimation (MLE)
in Positron Emission Tomography (PET) and propose a new algorithmic structure
called the StringAveraging ExpectationMaximization (SAEM). In our simulation
study, highcontrast and less noisy images with clear object boundaries are
reconstructed with the proposed SAEM 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 Lcurve 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.
20062009
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 JianGuo 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 YuChen 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 metaldielectric 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.
20012005
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 PoissonBoltzmann equation with discontinuous coe_ cients was
proposed. The method is secondorder and of linear complexity, whereas most
existing method is _rstorder. 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 SpringerVerlog[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, generalpurpose front tracking code for solving gas dynamic problems
such as shock di_raction problem, RayleighTaylor problem, RichtmyerMeshkov
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 controlvolume method on icosahedral
grid for the shallowwater 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.
 Nondivergent
barotropic equations on the sphere The nondivergent barotropic equations on a
sphere provide the simplest, yet the most important mathematical model for the
description of largescale horizontal motions of the atmosphere. Based on
Holder estimation, we prove global existence, uniqueness, and regularity
theorems for the nondivergent 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
densitystrati_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 dipolarvortex dynamics. This work also gave an
insight of the wellknown RichtmyerMeshkov instability in the mixing uid ow,
namely, the evolution of a shockaccelerated densitystrati_ed interface was
determined by (1) the vorticity deposition by the shockinterface 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 wellknown problem involving the global existence
of a solution for hyperbolic conservation laws with large data. In another work,
I proved the global existence for nongenuinely 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 timeasymptotic 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, \multiplemode di_usion waves"
were the timeasymptotic 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
nonequilibrium gas dynamics, ood ow with friction, viscoelasticility,
magnetohydrodynamics, etc. I have shown that the longrange e_ect of relaxation
is equivalent to a viscous e_ect. As a consequence, the longtime 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
waveinteraction picture near fronts.(cited 33)
 I identi_ed that
the major error source of the LaxFriedrichs scheme was the discrete di_usion
waves. They were also the timeasymptotic of the LaxFriedrichs _nite di_erence equations.
Construction and properties of these discrete di_usion were also given
[33] (cited 9).
