Skip to publications in [1992-1995] [1996] [1997] [1998] [1999] [ 2000] [2001] [2002] [2003] [2004]…[2009]
The Direct Discontinuous Galerkin (DDG) method for diffusion with interface corrections (with Jue Yan ), UCLA CAM report 09-44.
Kinetic models for polymers with inertial effects (with P. Degond, 2009), arXiv.0901.1909
Global regularity of the 4D Restricted Euler equations (with E. Tadmor and D. Wei, 2008) arXiv:0810.1964
Formulation and analysis of alternating evolution (AE) schemes for hyperbolic conservation laws (with H. Ahmed, 2008), submitted to SIAM J. on Sci. Computing.
Journal publications:
2009
[69] Tong Li and H. Liu
Critical thresholds in hyperbolic relaxation systems, J. Diff. Equ. 247(1) (2009), 33--48.
[67] H. Liu and Zhongming Wang
A Bloch band based level set method for computing the semi-classical limit of Schroedinger equations, J. Comp. Phys. 228(9) (2009), 3326--3344.
[66] Tong Li and H. Liu
Critical thresholds in a relaxation system with resonance of
characteristic speeds, Discrete and Continuous Dynamical Systems,
Series A. 24(2) (2009), 511-521.
[65] H. Liu and J.
Yan
The Direct
Discontinuous Galerkin (DDG) methods for diffusion
problems, SIAM Journal on Numerical Analysis, 47(1) (2009), 675--698.
2008
[64] H. Liu
On discreteness of the Hopf equation,
Acta Math. Appl. Sin. Engl. Ser.
24 (2008), no. 3, 423--440.
[63] H. Liu and Z.M. Wang
Superposition of multi-valued solutions in high
frequency wave dynamics, Journal of Scientific Computing, 35(2-3) (2008),
192--218.
[62] H. Liu and C. Sparber
Rigorous Derivation of the Hydrodynamical Equations for Rotating Superfluids, Mathematical Models and Methods in Applied Sciences, 18(5) (2008), 689—706.
[61] H. Liu
An alternating evolution approximation to systems of hyperbolic conservation
laws, Journal
of Hyperbolic Differential Equations, 5(2) (2008), 1--27.
[60] T. Li and H. Liu
Critical thresholds in a relaxation model for travel
flows, Indiana Univ. Math. Journal,
57(3) (2008), 1409--1430.
[59] M. D. Francesco, K. Fellner and H. Liu
A non-local conservation law with nonlinear `radiation’inhomogeneity.
J. Hyperbolic Differ. Equ. 5 (2008), no. 1, 1--23.
[58] C. Liu and H. Liu,
Boundary conditions for the microscopic FENE models, SIAM J. Appl. Math. 68 (2008), no. 5, 1304--1315.
[57] H. Liu
Global orientation dynamics for
liquid crystalline polymers, Physica D. 228 (2007), 122-129.
[56] H. Liu and Z.M. Wang,
A field space-based level set method for computing
multi-valued solutions to 1D Euler-Poisson equations,
Journal of Computational Physics, 225 (2007), 591--614.
[55] H. Liu and Z.M. Wang
Computing multi-valued velocity and electric fields
for 1D Euler-Poisson equations,
Applied Numerical Mathematics, 57 (2007), 821—836.
[54] H. Liu, S. Osher and R. Tsai,
Multi-valued solution and level set methods in computational high frequency wave propagation, Commun. Comput. Phys. 1(5) (2006), 765-804.
[53] H. Liu
Wave Breaking in a class of
nonlocal dispersive wave equations, Journal of
Nonlinear Math Phys. 13 (3), (2006), 441-466.
[52] H. Liu and J. Yan
A local discontinuous {G}alerkin method for the {K}orteweg-de
{V}ries equation with boundary effect, JCP
215 (2006), 197-218.
[51] H. Liu
Critical Thresholds in the Semiclassical
Limit of 2-D
Rotational Schr\"{o}dinger Equations, ZAMP.
57 (2006), 42-58.
[50] H. Liu, H. Zhang and P.W. Zhang
Axial Symmetry and Classification of
Stationary Solutions of
Doi-Onsager Equation on the Sphere with Maier-Saupe Potential, Comm. Math. Sci. Vol 3 (2), (2005), 201-218.
[49] S. Jin, H.L. Liu, S. Osher and R. Tsai
Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems, JCP, 210(2005), 498-518.
[48] S. Jin, H.L. Liu, S. Osher and R. Tsai
(2005)
Computing multi-valued physical observables for the semiclassical limit of the Schroedinger equation, JCP, 205 (2005), 222-241.
[47] T. Li and
H. Liu
Stability of a traffic flow model with nonconvex relaxation, Comm. Math. Sci., Vol 3 (2), (2005), 101--118.
[46] H. Liu
Relaxation
Dynamics, Scaling Limits and Convergence of Relaxation Schemes, (2005) Springer Verlag.
[45] H.L. Liu and E. Tadmor
Rotation
Prevents Finite Time Breakdown, Physica
D 188 (2004) 262-276.
[44]G.-Q. Chen and H.L. Liu
Concentration
and Cavitation in Solutions of the Euler equations
for nonisentropic fluids as the pressure vanishes Physics.
D. 189 (2004), 141--165.
[43] H.L. Liu and M. Slemrod
KdV Dynamics in the
Plasma-Sheath Transition, Appl. Math. Lett. 17 (
2004) 401--419.
[42] S. Jin, H.L. Liu, S. Osher and R. Tsai
Computing
multivalued physical observables for the semiclassical limit of the Schrodinger equations,
JCP, 205 (1) (2005), 222-241.
[41] H.L. Liu, L.-T. Cheng and S. Osher (2003)
A level set framework for capturing multi-valued
solutions to nonlinear first-order equations, to appear in J. Sci. Comp.
[40] L.T. Cheng, H.L. Liu and S. Osher
Computational
high frequency wave propagation using the level set method,
with applications to the semi-classical limit
of Schroedinger equations, Comm. Math. Sci. Vol 1, No. 3 (2003), 593--621.
[39] L.T. Cheng, H.L. Liu and S. Osher
High frequency wave propagation in Schroedinger
equations using the Level Set method, preprint, 2003.
[38] H.L. Liu and E. Tadmor
Critical
Thresholds in 2-D Restricted Euler-Poisson Equations, SIAM
J. Appl. Math. 63 (6) (2003), 1889--1910.
[37] H.L. Liu
Asymptotic
Stability of Relaxation Shock Profiles for Hyperbolic Conservation Laws,
J. Diff. Equ. 192 (2003), 285--307.
[36] G.-Q. Chen and H.L. Liu
Formation
of Delta-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions
to the Isentropic Euler Equations
SIAM J. Math Anal. 34 (2003), 925--938.
[35] J. Pan and H.L. Liu
Convergence
rates to traveling waves of viscous conservation laws with dispersion
J. Diff. Equ. 187 (2003), 337--358.
[34] H. L. Liu
The l^1 Global Decay to Discrete Shocks for Scalar Monotone
Schemes, Math. Comp. 72
(2003), 227-245.
2002
[33] H.L. Liu and E. Tadmor
Semi
classical Limit of the Nonlinear Schrodinger-Poisson Equation
with Subcritical Initial Data
Methods and Applications of Analysis, Vol 9,
No. 4 (2002), 517--532.
[32] H.L. Liu and E. Tadmor (2002)
Critical
Thresholds and Conditional Stability for Euler Equations and Related Models,
Proceedings
of the Ninth International Conference on ''Hyperbolic Problems: Theory, Numerics, Applications",
Editors: T.Y. Hou and E. Tadmor, Springer, pp227--240.
[31] H.L. Liu and
Spectral
Dynamics of the Velocity Gradient Field in Restricted Fluid Flows
Commun. Math. Phys. 228 (2002), 435--466.
[30] H.L. Liu and E. Tadmor
Critical
Thresholds in a Convolution Model for Nonlinear Conservation Laws,
SIAM J. Math. Anal. 33 (2002), 930--945.
[29] H.T. Fan and H.L. Liu (2001)
Pattern Formation,
Wave Propagation and Stability in Conservation Laws with Slow
Diffusion and Fast Reaction
UCLA CAM report 01-24. submitted
[28] S. Engelberg, H.L. Liu and
Critical
Thresholds in Euler-Poisson equations,
Indiana University Mathematics Journal, 50 (2001), 109--157.
[27] H.L. Liu and W.-A. Yong
Time-Asymptotic stability of boundary-layers for a hyperbolic
relaxation system, ,
Comm. Partial Diff. Equ., 26 (2001),
1323--1343.
[26] H.L. Liu
Asymptotic
decay to the relaxation shock fronts in two dimensions,
Proceedings of the Royal Society of Edinburgh: Section A 131A
(2001), 1385--1410.
[25] H. L. Liu, J. Wang and G. Warnecke
Convergence
of a splitting scheme applied to the R-W model of the Boltzmann equation, J. Comp. Appl. Math. 134 (2001), 343--367.
[24] H.L. Liu, J. Wang and G. Warnecke
Convergence
rate to discrete shocks for nonconvex conservations,
Numerische Mathematik,
88 (2001), 513-541.
[23] H. Liu and R. Natalini
Long-Time
Diffusive Behavior of Solutions to a Hyperbolic Relaxation System, Asymptotic Analysis, 25 (2001) 21-38.
[22] H.L. Liu
The L^p Stability of Relaxation Rarefaction Profiles, J. Differ Equations, 171, (2001),
397-411.
2000
[21] H. Liu, J. Wang and G. Warnecke
The
lip^+ Stability and Error Estimates for a Relaxation Scheme, SIAM J. Numer. Anal. 38
(2000), no. 4, 1154--1170
[20] S. Jin and H. L. Liu
A
Diffusive Subcharacteristic Condition for Hyperbolic
Systems with Diffusive Relaxation, Transport Theory and Statistical
Physics, 29 (3-5), (2000), 583--593.
[19] H.L. Liu, G. Warnecke
Convergence
rates for relaxation schemes approximating conservation laws, SIAM J. Numer. Anal. 37, No. 4, (2000), 1316--1337.
[18] H.L. Liu
Convergence
rate to the discrete traveling wave for relaxation schemes, Math.
Comp. 69 (2000), 583--608.
1999
[17]H.L. Liu, J. Wang and T. Yang
Nonlinear stability and existence of stationary discrete traveling waves for the
relaxing schemes, Japan J. Indust. Appl. Math. 16
(1999), 195--224.
1998
[16] J. Pan, H.L. Liu
On stability of travelling waves of Burgers-Fisher equation, Ann. Differential
Equations, 14 (1998), 37--47.
[15] H.L. Liu
Nonlinear stability of shock profiles for non-convex model equations with
degenerate shock, J. Partial Differential Equations 11(1998), 209--230.
[14] S. Jin and H. L. Liu
Diffusion limit of a hyperbolic system with relaxation, Methods Appl. Anal.
5 (1998) 317-334.
[13] H.L. Liu, J. Wang and T. Yang
Stability
for a relaxation model with a nonconvex flux,
SIAM J. Math. Anal. 29 (1998), 18-29.
[12] H.L. Liu and J. Wang
Asymptotic stability of stationary discrete shocks of Lax-Friedrichs
scheme for non-convex conservation laws, Japan J. Indust.
Appl. Math. 15 (1998), 1-16.
1997
[11] H.L. Liu, C. W. Woo and T. Yang
Decay rate for traveling waves of a relaxation model, J. Diff. Equ. 134 (1997), 343-367.
[10]H. L. Liu, J. Wang and T. Yang
Existence of the discrete traveling waves for a relaxing scheme, Appl. Math. lett., 10 (1997), 117-122.
[9]H.L. Liu
Asymptotic stability of shock profiles for non-convex convection diffusion
equation, Appl. Math. Lett. 10 (1997),
129-134.
[8] H.L. Liu, J. Wang
Decay rate for perturbations of stationary discrete shocks for convex
scalar conservation laws, Math. Comp. 66 (1997), 69-84.
1996
[7] H. L. Liu
Asymptotic properties of solutions to nonconvex
hyperbolic conservation laws.(Chinese), Gaoxiao Yingyong Shuxue Xuebao Ser. A 11 (1996), 277--282.
[6] H.L. Liu, J. Wang
Asymptotic stability of traveling wave solution for a hyperbolic system with
relaxation terms, Beijing Math. 2 (1996), 119-130.
[5] H. L. Liu, J. Wang
Nonlinear stability of stationary discrete profiles of non-convex scalar
conservation laws, Math. Comp. 65 (1996), 1137-1153.
1992-1995
[4] H. Liu and J. Pan
Decay rate for perturbations of viscous shock profiles for non-convex
convection-diffusion equation, Appl. Functional Anal. 2 (1995), 171-176.
[3] H. L. Liu
An existence theorem for radial positive solutions of nonlinear elliptic
equations, Sys. Sci. Math. Sci. 7 (1994), 1-4.
[2] H. L. Liu
The interactions of shock waves of nonstrictly
hyperbolic systems, Acta Math. Scientia,
12 (1992), 312-336.
[1] H. L. Liu
Large time behavior of solutions of the porous medium equation with convection,
Acta Mathematicae Appl. Sinica 15 (2) (1992), 239-256.