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## Liu, Hailiang

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 Author ID: liu.hailiang Published as: Liu, Hailiang; Liu, H.; Liu, Hai-liang; Liu, H. L.
 Documents Indexed: 185 Publications since 1976
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#### Co-Authors

 24 single-authored 9 Wang, Jinghua 8 Tadmor, Eitan 8 Wang, Zhongming 7 Huang, Yunqing 6 Warnecke, Gerald G. 6 Yi, Nianyu 5 Li, Tong 5 Osher, Stanley Joel 5 Yin, Peimeng 5 Yu, Hui 4 Cai, Wenli 4 Jin, Shi 4 Ralston, James V. 4 Yang, Tong 3 Jabin, Pierre-Emmanuel 3 Jiang, Yi 3 Lee, Yongki 3 Pollack, Michael 3 Runborg, Olof 3 Tanushev, Nicolay M. 3 Tsai, Yen-Hsi Richard 3 Yan, Jue 2 Bernussou, Jacques 2 Bhatnagar, Manas 2 Chen, Gui-Qiang G. 2 Cheng, Li-Tien 2 Degond, Pierre 2 Hadji, Layachi 2 Himeno, Ryutaro 2 Krishnan, Sreedevi 2 Lu, Wenying 2 Marella, Saikrishna V. 2 Pan, Jun 2 Pryporov, Maksym 2 Saran, Haseena 2 Schreiber, Willard C. 2 Shin, Jaemin 2 Udaykumar, Holavanahalli S. 2 Wen, Hairui 2 Yamaguchi, Takami 1 Akulichev, Victor A. 1 Aten, Charlotte 1 Ates, Muzaffer 1 Ayachi, B. 1 Bajcinca, Naim 1 Bau, E. 1 Bazargan-Sabet, B. 1 Butterfaß, J. 1 Cao, Waixiang 1 Cesnik, Carlos E. S. 1 Chanchole, S. 1 Chui, E. 1 Davulcu, Hasan 1 Di Francesco, Marco 1 Dirr, Bodo 1 Engelberg, Shlomo 1 Fan, Haitao 1 Fellner, Klemens 1 Fox, Rodney O. 1 Garg, Hari Krishna 1 Gröpler, Robin 1 Ha, Tae Gab 1 Ifju, Peter G. 1 Ikehata, Masaru 1 Iosevich, Alex 1 Jackiewicz, Zdzislaw 1 Knoch, S. 1 Kolippakkam, D. 1 Kuang, Yang 1 Lebon, Frédéric 1 Levermore, C. David 1 Li, Bingtuan 1 Lian, Yongsheng 1 Liao, Guojun 1 Lien, Fue-Sang 1 Liu, Chun 1 Lott, Adam 1 MacKinnon, I. 1 Maimaitiyiming, Wumaier 1 Maimon, Shir 1 Markowich, Peter Alexander 1 McCartney, C. J. 1 Mehl, Michael J. 1 Meusel, P. 1 Mira, Christian 1 Mu, S. Y. 1 Natalini, Roberto 1 Nguyen, Hung V. 1 Nonami, Kenzo 1 Papaconstantopoulos, D. A. 1 Pego, Robert L. 1 Pendleton, Terrance 1 Petridis, Giorgis 1 Ploymaklam, Nattapol 1 Popp, Karl Karl 1 Qian, Linrui 1 Qin, Yuming 1 Ramière, Isabelle 1 Rojas Mena, C. 1 Savelief, D. ...and 23 more Co-Authors
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#### Serials

 21 Journal of Computational Physics 8 Mathematics of Computation 8 Journal of Differential Equations 8 Journal of Scientific Computing 6 SIAM Journal on Scientific Computing 5 SIAM Journal on Numerical Analysis 5 Physica D 5 Applied Mathematics Letters 5 SIAM Journal on Mathematical Analysis 4 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 4 Communications in Mathematical Sciences 4 Communications in Computational Physics 3 International Journal for Numerical Methods in Fluids 3 International Journal of Solids and Structures 3 Journal of Computational and Applied Mathematics 3 Numerische Mathematik 3 Numerical Methods for Partial Differential Equations 3 SIAM Journal on Applied Mathematics 3 Methods and Applications of Analysis 3 Discrete and Continuous Dynamical Systems 3 Journal of Hyperbolic Differential Equations 2 Journal of Mathematical Analysis and Applications 2 Molecular Simulation 2 Indiana University Mathematics Journal 2 Acta Mathematica Hungarica 2 Japan Journal of Industrial and Applied Mathematics 2 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2 Multiscale Modeling & Simulation 2 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 2 Advances and Applications in Fluid Mechanics 1 Communications in Mathematical Physics 1 Computer Methods in Applied Mechanics and Engineering 1 Computers and Structures 1 IMA Journal of Numerical Analysis 1 International Journal of Control 1 International Journal of Non-Linear Mechanics 1 International Journal of Systems Science 1 Journal of Sound and Vibration 1 Nonlinearity 1 Transport Theory and Statistical Physics 1 Wave Motion 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 1 Fuzzy Sets and Systems 1 International Journal for Numerical Methods in Engineering 1 Journal of Functional Analysis 1 Mathematics and Computers in Simulation 1 Mechanics Research Communications 1 Acta Mathematicae Applicatae Sinica 1 Circuits, Systems, and Signal Processing 1 Parallel Computing 1 Applied Numerical Mathematics 1 Acta Mathematicae Applicatae Sinica. English Series 1 Annals of Differential Equations 1 Asymptotic Analysis 1 Signal Processing 1 Systems Science and Mathematical Sciences 1 Communications in Partial Differential Equations 1 Communications in Statistics. Simulation and Computation 1 Acta Mathematica Scientia 1 Journal of Partial Differential Equations 1 Advances in Engineering Software 1 Applied Mathematics. Series A (Chinese Edition) 1 Acustica - Acta Acustica 1 Journal of Vibration and Control 1 Mechanism and Machine Theory 1 Flow, Turbulence and Combustion 1 Journal of Nonlinear Mathematical Physics 1 Communications in Information and Systems 1 Quantum Information Processing 1 Iranian Journal of Fuzzy Systems 1 Acta Mechanica Sinica 1 Networks and Heterogeneous Media 1 Journal of Nonlinear Science and Applications 1 Quantum Studies: Mathematics and Foundations
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#### Fields

 120 Partial differential equations (35-XX) 76 Numerical analysis (65-XX) 42 Fluid mechanics (76-XX) 20 Statistical mechanics, structure of matter (82-XX) 15 Biology and other natural sciences (92-XX) 12 Mechanics of deformable solids (74-XX) 7 Quantum theory (81-XX) 5 Mechanics of particles and systems (70-XX) 5 Optics, electromagnetic theory (78-XX) 5 Operations research, mathematical programming (90-XX) 4 Dynamical systems and ergodic theory (37-XX) 4 Approximations and expansions (41-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 3 Ordinary differential equations (34-XX) 3 Integral equations (45-XX) 3 Computer science (68-XX) 3 Geophysics (86-XX) 3 Systems theory; control (93-XX) 2 Number theory (11-XX) 2 Difference and functional equations (39-XX) 2 Convex and discrete geometry (52-XX) 2 Differential geometry (53-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Statistics (62-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Information and communication theory, circuits (94-XX) 1 Combinatorics (05-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Group theory and generalizations (20-XX) 1 Abstract harmonic analysis (43-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)

#### Citations contained in zbMATH Open

136 Publications have been cited 1,510 times in 686 Documents Cited by Year
Formation of $$\delta$$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. Zbl 1038.35035
Chen, Gui-Qiang; Liu, Hailiang
2003
Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Zbl 1098.76603
Chen, Gui-Qiang; Liu, Hailiang
2004
The direct discontinuous Galerkin (DDG) methods for diffusion problems. Zbl 1189.65227
Liu, Hailiang; Yan, Jue
2009
Sharp interface Cartesian grid method. I: An easily implemented technique for 3D moving boundary computations. Zbl 1154.76359
Marella, S.; Krishnan, S.; Liu, H.; Udaykumar, H. S.
2005
The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Zbl 1364.65200
Liu, Hailiang; Yan, Jue
2010
Critical thresholds in Euler-Poisson equations. Zbl 0989.35110
Engelberg, Shlomo; Liu, Hailiang; Tadmor, Eitan
2001
Computational high-frequency wave propagation using the level set method with applications to the semi-classical limit of Schrödinger equations. Zbl 1084.35066
Cheng, Li-Tien; Liu, Hailiang; Osher, Stanley
2003
Critical thresholds in a convolution model for nonlinear conservation laws. Zbl 1002.35085
2001
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation. Zbl 1072.65132
Jin, Shi; Liu, Hailiang; Osher, Stanley; Tsai, Yen-Hsi Richard
2005
Spectral dynamics of the velocity gradient field in restricted flows. Zbl 1031.76006
2002
A numerical study of insect flight. Zbl 0929.76092
Liu, H.; Kawachi, K.
1998
Sharp interface Cartesian grid method. II: A technique for simulating droplet interactions with surfaces of arbitrary shape. Zbl 1154.76358
Liu, H.; Krishnan, S.; Marella, S.; Udaykumar, H. S.
2005
Effect of initial stress on the propagation behavior of Love waves in a layered piezoelectric structure. Zbl 1005.74033
Liu, H.; Wang, Z. K.; Wang, T. J.
2001
A numerical study of undulatory swimming. Zbl 0958.76099
Liu, H.; Kawachi, K.
1999
Critical thresholds in 2D restricted Euler-Poisson equations. Zbl 1073.35187
2003
A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect. Zbl 1092.65083
Liu, Hailiang; Yan, Jue
2006
Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential. Zbl 1092.76007
Liu, Hailiang; Zhang, Hui; Zhang, Pingwen
2005
Wave breaking in a class of nonlocal dispersive wave equations. Zbl 1110.35069
Liu, Hailiang
2006
A free energy satisfying finite difference method for Poisson-Nernst-Planck equations. Zbl 1349.65317
Liu, Hailiang; Wang, Zhongming
2014
A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: energy conservation and boundary effect. Zbl 1297.65122
Yi, Nianyu; Huang, Yunqing; Liu, Hailiang
2013
Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations. Zbl 1331.65141
Liu, Hailiang
2015
Rotation prevents finite-time breakdown. Zbl 1098.76643
2004
Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. Zbl 1098.65094
Jin, Shi; Liu, Hailiang; Osher, Stanley; Tsai, Richard
2005
Error estimates for Gaussian beam superpositions. Zbl 1277.35236
Liu, Hailiang; Runborg, Olof; Tanushev, Nicolay M.
2013
The entropy satisfying discontinuous Galerkin method for Fokker-Planck equations. Zbl 1320.65143
Liu, Hailiang; Yu, Hui
2015
Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations. Zbl 1361.65061
Cao, Waixiang; Liu, Hailiang; Zhang, Zhimin
2017
Computational aerodynamics of low Reynolds number plunging, pitching and flexible wings for MAV applications. Zbl 1271.76285
Shyy, W.; Lian, Y.; Tang, J.; Liu, H.; Trizila, P.; Stanford, B.; Bernal, L.; Cesnik, C.; Friedmann, P.; Ifju, P.
2008
A local discontinuous Galerkin method for the Burgers-Poisson equation. Zbl 1339.65177
Liu, Hailiang; Ploymaklam, Nattapol
2015
Mass preserving discontinuous Galerkin methods for Schrödinger equations. Zbl 1352.65358
Lu, Wenying; Huang, Yunqing; Liu, Hailiang
2015
Multi-valued solution and level set methods in computational high frequency wave propagation. Zbl 1120.65110
Liu, H.; Osher, S.; Tsai, R.
2006
Entropy-based fuzzy clustering and fuzzy modeling. Zbl 1147.62348
Yao, J.; Dash, M.; Tan, S. T.; Liu, H.
2000
Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws. Zbl 1039.35063
Liu, Hailiang
2003
Recovery of high frequency wave fields from phase space-based measurements. Zbl 1221.35031
Liu, Hailiang; Ralston, James
2010
Maximum-principle-satisfying third order discontinuous Galerkin schemes for Fokker-Planck equations. Zbl 1341.65038
Liu, Hailiang; Yu, Hui
2014
Stability of a relaxation model with a nonconvex flux. Zbl 0961.35017
Liu, Hailiang; Wang, Jinghua; Yang, Tong
1998
Kinetic models for polymers with inertial effects. Zbl 1183.76614
Degond, Pierre; Liu, Hailiang
2009
A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson-Nernst-Planck systems. Zbl 1406.82021
Liu, Hailiang; Wang, Zhongming
2017
An entropy satisfying discontinuous Galerkin method for nonlinear Fokker-Planck equations. Zbl 1354.35158
Liu, Hailiang; Wang, Zhongming
2016
A note on harmonic maps. Zbl 0996.58507
Liu, H.; Liao, Guojun
1996
Decay rate for travelling waves of a relaxation model. Zbl 0908.35074
Liu, Hailiang; Woo, Ching Wah; Yang, Tong
1997
Convergence rates for relaxation schemes approximating conservation laws. Zbl 0954.35108
Liu, Hailiang; Warnecke, Gerald
2000
Surfaces in the lightlike cone. Zbl 1108.53011
Liu, H. L.
2007
Recovery of normal derivatives from the piecewise $$L^2$$ projection. Zbl 1408.65009
Huang, Yunqing; Liu, Hailiang; Yi, Nianyu
2012
On discreteness of the Hopf equation. Zbl 1149.65071
Liu, Hai-liang
2008
A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation. Zbl 1349.65462
Liu, Hailiang; Yi, Nianyu
2016
KdV dynamics in the plasma-sheath transition. Zbl 1060.76137
Liu, Hailiang; Slemrod, M.
2004
Recovery of high frequency wave fields for the acoustic wave equation. Zbl 1213.35284
Liu, Hailiang; Ralston, James
2010
Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Zbl 1304.35420
Lee, Yongki; Liu, Hailiang
2015
An invariant preserving discontinuous Galerkin method for the Camassa-Holm equation. Zbl 1382.65321
Liu, Hailiang; Xing, Yulong
2016
The $$L^p$$ stability of relaxation rarefaction profiles. Zbl 0988.35113
Liu, Hailiang
2001
Critical thresholds in hyperbolic relaxation systems. Zbl 1170.35062
Li, Tong; Liu, Hailiang
2009
An entropy satisfying conservative method for the Fokker-Planck equation of the finitely extensible nonlinear elastic dumbbell model. Zbl 1422.74097
Liu, Hailiang; Yu, Hui
2012
A level set framework for capturing multi-valued solutions of nonlinear first-order equations. Zbl 1109.65080
Liu, Hailiang; Cheng, Li-Tien; Osher, Stanley
2006
Stability of a traffic flow model with nonconvex relaxation. Zbl 1099.35156
Li, Tong; Liu, Hailiang
2005
Semiclassical limit of the nonlinear Schrödinger-Poisson equation with subcritical initial data. Zbl 1166.35374
2002
Alternating evolution discontinuous Galerkin methods for Hamilton-Jacobi equations. Zbl 1349.65461
Liu, Hailiang; Pollack, Michael
2014
Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations. Zbl 1218.35175
Liu, Hailiang; Yin, Zhaoyang
2010
Global regularity of the 4D restricted Euler equations. Zbl 1193.37123
Liu, Hailiang; Tadmor, Eitan; Wei, Dongming
2010
Robust control of robotic manipulators based on integral sliding mode. Zbl 1154.93344
Shi, J.; Liu, H.; Bajcinca, N.
2008
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit. Zbl 1395.76047
Degond, P.; Liu, H.; Savelief, D.; Vignal, M.-H.
2012
Global well-posedness for the microscopic FENE model with a sharp boundary condition. Zbl 1233.35109
Liu, Hailiang; Shin, Jaemin
2012
Boundary conditions for the microscopic FENE models. Zbl 1147.76003
Liu, Chun; Liu, Hailiang
2008
Diffusion limit of a hyperbolic system with relaxation. Zbl 0922.35086
Jin, Shi; Liu, Hailiang
1998
Computing multi-valued velocity and electric fields for 1D Euler-Poisson equations. Zbl 1146.78012
Liu, Hailiang; Wang, Zhongming
2007
Superposition of multi-valued solutions in high frequency wave dynamics. Zbl 1203.65141
Liu, Hailiang; Wang, Zhongming
2008
Alternating evolution discontinuous Galerkin methods for convection-diffusion equations. Zbl 1352.65356
Liu, Hailiang; Pollack, Michael
2016
Semi-active fuzzy sliding mode control of full vehicle and suspensions. Zbl 1182.70067
Liu, H.; Nonami, K.; Hagiwara, T.
2005
A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems. Zbl 1407.65168
Liu, Hailiang; Yin, Peimeng
2018
A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations. Zbl 1140.81399
Liu, Hailiang; Wang, Zhongming
2007
Alternating evolution schemes for hyperbolic conservation laws. Zbl 1248.65092
Saran, Haseena; Liu, Hailiang
2011
Tiling sets and spectral sets over finite fields. Zbl 1376.52038
Aten, C.; Ayachi, B.; Bau, E.; FitzPatrick, D.; Iosevich, A.; Liu, H.; Lott, A.; MacKinnon, I.; Maimon, S.; Nan, S.; Pakianathan, J.; Petridis, G.; Rojas Mena, C.; Sheikh, A.; Tribone, T.; Weill, J.; Yu, C.
2017
A nonlocal conservation law with nonlinear “radiation” inhomogeneity. Zbl 1158.35063
di Francesco, Marco; Fellner, Klemens; Liu, Hailiang
2008
Global orientation dynamics for liquid crystalline polymers. Zbl 1183.82101
Liu, Hailiang
2007
Critical thresholds in the semiclassical limit of 2D rotational Schrödinger equations. Zbl 1089.35052
Liu, Hailiang
2006
An iterative discontinuous Galerkin method for solving the nonlinear Poisson Boltzmann equation. Zbl 1388.65162
Yin, Peimeng; Huang, Yunqing; Liu, Hailiang
2014
Gaussian beam methods for the Helmholtz equation. Zbl 1302.35135
Liu, Hailiang; Ralston, James; Runborg, Olof; Tanushev, Nicolay M.
2014
Global dynamics of Bose-Einstein condensation for a model of the Kompaneets equation. Zbl 1362.37161
Levermore, C. David; Liu, Hailiang; Pego, Robert L.
2016
Time-asymptotic stability of boundary-layers for a hyperbolic relaxation system. Zbl 0997.35007
Liu, Hailiang; Yong, Wen-An
2001
Nonlinear stability and existence of stationary discrete travelling waves for the relaxing schemes. Zbl 0934.65101
Liu, Hailiang; Wang, Jinghua; Yang, Tong
1999
Long-time diffusive behavior of solutions to a hyperbolic relaxation system. Zbl 0981.35035
Liu, Hailiang; Natalini, Roberto
2001
Global attractor for a nonlinear thermoviscoelastic system in shape memory alloys. Zbl 1173.35386
Qin, Yuming; Liu, Hailiang; Song, Changming
2008
Critical thresholds in a relaxation model for traffic flows. Zbl 1152.35006
Li, Tong; Liu, Hailiang
2008
Error estimates for the AEDG method to one-dimensional linear convection-diffusion equations. Zbl 1376.65122
Liu, Hailiang; Wen, Hairui
2018
A conservative discontinuous Galerkin method for the Degasperis-Procesi equation. Zbl 1296.35158
Huang, Yunqing; Liu, Hailiang; Yi, Nianyu
2014
On traveling wave solutions of the $$\theta$$-equation of dispersive type. Zbl 1304.35190
Ha, Tae Gab; Liu, Hailiang
2015
Time-asymptotic convergence rates towards the discrete evolutionary stable distribution. Zbl 1328.65268
Cai, Wenli; Jabin, Pierre-Emmanuel; Liu, Hailiang
2015
A uniform multiscale method for 2D static and dynamic analyses of heterogeneous materials. Zbl 1352.74459
Zhang, H. W.; Liu, H.; Wu, J. K.
2013
Asymptotic stability of shock profiles for nonconvex convection-diffusion equation. Zbl 0888.35052
Liu, Hailiang
1997
Computation of free surface waves around an arbitrary body by a Navier- Stokes solver using the pseudocompressibility technique. Zbl 0814.76066
Liu, H.; Ikehata, M.
1994
Pattern formation, wave propagation and stability in conservation laws with slow diffusion and fast reaction. Zbl 1065.35156
Fan, Haitao; Liu, Hailiang
2004
Critical thresholds in one-dimensional damped Euler-Poisson systems. Zbl 1453.35145
Bhatnagar, Manas; Liu, Hailiang
2020
Critical thresholds in a relaxation system with resonance of characteristic speeds. Zbl 1171.35428
Li, Tong; Liu, Hailiang
2009
Threshold for shock formation in the hyperbolic Keller-Segel model. Zbl 1330.35250
Lee, Yongki; Liu, Hailiang
2015
An alternating evolution approximation to systems of hyperbolic conservation laws. Zbl 1162.35049
Liu, Hailiang
2008
Unconditionally energy stable DG schemes for the Swift-Hohenberg equation. Zbl 1427.65372
Liu, Hailiang; Yin, Peimeng
2019
Effect of a biasing electric field on the propagation of antisymmetric Lamb waves in piezoelectric plates. Zbl 1036.74027
Liu, H.; Wang, T. J.; Wang, Z. K.; Kuang, Z. B.
2002
Convergence rates to traveling waves for viscous conservation laws with dispersion. Zbl 1019.35018
Pan, Jun; Liu, Hailiang
2003
Relaxation dynamics, scaling limits and convergence of relaxation schemes. Zbl 1094.35078
Liu, Hailiang
2005
On invariant-preserving finite difference schemes for the Camassa-Holm equation and the two-component Camassa-Holm system. Zbl 1388.65059
Liu, Hailiang; Pendleton, Terrance
2016
On a non-local selection-mutation model with a gradient flow structure. Zbl 1382.35313
Jabin, Pierre-Emmanuel; Liu, Hailiang
2017
Efficient, positive, and energy stable schemes for multi-d Poisson-Nernst-Planck systems. Zbl 07358941
Liu, Hailiang; Maimaitiyiming, Wumaier
2021
Critical thresholds in one-dimensional damped Euler-Poisson systems. Zbl 1453.35145
Bhatnagar, Manas; Liu, Hailiang
2020
Unconditionally energy stable DG schemes for the Swift-Hohenberg equation. Zbl 1427.65372
Liu, Hailiang; Yin, Peimeng
2019
Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity. Zbl 1452.65260
Yu, Hui; Liu, Hailiang
2019
A conservative discontinuous Galerkin method for nonlinear electromagnetic Schrödinger equations. Zbl 1435.35364
Yi, Nianyu; Huang, Yunqing; Liu, Hailiang
2019
On accuracy of the mass-preserving DG method to multi-dimensional Schrödinger equations. Zbl 1461.65241
Liu, Hailiang; Huang, Yunqing; Lu, Wenying; Yi, Nianyu
2019
A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems. Zbl 1407.65168
Liu, Hailiang; Yin, Peimeng
2018
Error estimates for the AEDG method to one-dimensional linear convection-diffusion equations. Zbl 1376.65122
Liu, Hailiang; Wen, Hairui
2018
Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations. Zbl 1416.65293
Jiang, Yi; Liu, Hailiang
2018
An invariant-region-preserving (IRP) limiter to DG methods for compressible Euler equations. Zbl 1407.65191
Jiang, Yi; Liu, Hailiang
2018
Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations. Zbl 1361.65061
Cao, Waixiang; Liu, Hailiang; Zhang, Zhimin
2017
A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson-Nernst-Planck systems. Zbl 1406.82021
Liu, Hailiang; Wang, Zhongming
2017
Tiling sets and spectral sets over finite fields. Zbl 1376.52038
Aten, C.; Ayachi, B.; Bau, E.; FitzPatrick, D.; Iosevich, A.; Liu, H.; Lott, A.; MacKinnon, I.; Maimon, S.; Nan, S.; Pakianathan, J.; Petridis, G.; Rojas Mena, C.; Sheikh, A.; Tribone, T.; Weill, J.; Yu, C.
2017
On a non-local selection-mutation model with a gradient flow structure. Zbl 1382.35313
Jabin, Pierre-Emmanuel; Liu, Hailiang
2017
A finite volume method for nonlocal competition-mutation equations with a gradient flow structure. Zbl 1378.92047
Cai, Wenli; Liu, Hailiang
2017
Error estimates for Gaussian beam methods applied to symmetric strictly hyperbolic systems. Zbl 07213155
Liu, Hailiang; Pryporov, Maksym
2017
An entropy satisfying discontinuous Galerkin method for nonlinear Fokker-Planck equations. Zbl 1354.35158
Liu, Hailiang; Wang, Zhongming
2016
A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation. Zbl 1349.65462
Liu, Hailiang; Yi, Nianyu
2016
An invariant preserving discontinuous Galerkin method for the Camassa-Holm equation. Zbl 1382.65321
Liu, Hailiang; Xing, Yulong
2016
Alternating evolution discontinuous Galerkin methods for convection-diffusion equations. Zbl 1352.65356
Liu, Hailiang; Pollack, Michael
2016
Global dynamics of Bose-Einstein condensation for a model of the Kompaneets equation. Zbl 1362.37161
Levermore, C. David; Liu, Hailiang; Pego, Robert L.
2016
On invariant-preserving finite difference schemes for the Camassa-Holm equation and the two-component Camassa-Holm system. Zbl 1388.65059
Liu, Hailiang; Pendleton, Terrance
2016
Sobolev and max norm error estimates for Gaussian beam superpositions. Zbl 1372.35263
Liu, Hailiang; Runborg, Olof; Tanushev, Nicolay M.
2016
Lightcone dual surfaces and hyperbolic dual surfaces of spacelike curves in de Sitter 3-space. Zbl 1346.53022
Liu, H.; Pei, D.
2016
Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations. Zbl 1331.65141
Liu, Hailiang
2015
The entropy satisfying discontinuous Galerkin method for Fokker-Planck equations. Zbl 1320.65143
Liu, Hailiang; Yu, Hui
2015
A local discontinuous Galerkin method for the Burgers-Poisson equation. Zbl 1339.65177
Liu, Hailiang; Ploymaklam, Nattapol
2015
Mass preserving discontinuous Galerkin methods for Schrödinger equations. Zbl 1352.65358
Lu, Wenying; Huang, Yunqing; Liu, Hailiang
2015
Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Zbl 1304.35420
Lee, Yongki; Liu, Hailiang
2015
On traveling wave solutions of the $$\theta$$-equation of dispersive type. Zbl 1304.35190
Ha, Tae Gab; Liu, Hailiang
2015
Time-asymptotic convergence rates towards the discrete evolutionary stable distribution. Zbl 1328.65268
Cai, Wenli; Jabin, Pierre-Emmanuel; Liu, Hailiang
2015
Threshold for shock formation in the hyperbolic Keller-Segel model. Zbl 1330.35250
Lee, Yongki; Liu, Hailiang
2015
Entropy satisfying schemes for computing selection dynamics in competitive interactions. Zbl 1318.65085
Liu, Hailiang; Cai, Wenli; Su, Ning
2015
Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation. Zbl 1348.35213
Liu, Hailiang; Pryporov, Maksym
2015
A free energy satisfying finite difference method for Poisson-Nernst-Planck equations. Zbl 1349.65317
Liu, Hailiang; Wang, Zhongming
2014
Maximum-principle-satisfying third order discontinuous Galerkin schemes for Fokker-Planck equations. Zbl 1341.65038
Liu, Hailiang; Yu, Hui
2014
Alternating evolution discontinuous Galerkin methods for Hamilton-Jacobi equations. Zbl 1349.65461
Liu, Hailiang; Pollack, Michael
2014
An iterative discontinuous Galerkin method for solving the nonlinear Poisson Boltzmann equation. Zbl 1388.65162
Yin, Peimeng; Huang, Yunqing; Liu, Hailiang
2014
Gaussian beam methods for the Helmholtz equation. Zbl 1302.35135
Liu, Hailiang; Ralston, James; Runborg, Olof; Tanushev, Nicolay M.
2014
A conservative discontinuous Galerkin method for the Degasperis-Procesi equation. Zbl 1296.35158
Huang, Yunqing; Liu, Hailiang; Yi, Nianyu
2014
Entropy/energy stable schemes for evolutionary dispersal models. Zbl 1349.65318
Liu, Hailiang; Yu, Hui
2014
A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: energy conservation and boundary effect. Zbl 1297.65122
Yi, Nianyu; Huang, Yunqing; Liu, Hailiang
2013
Error estimates for Gaussian beam superpositions. Zbl 1277.35236
Liu, Hailiang; Runborg, Olof; Tanushev, Nicolay M.
2013
A uniform multiscale method for 2D static and dynamic analyses of heterogeneous materials. Zbl 1352.74459
Zhang, H. W.; Liu, H.; Wu, J. K.
2013
Thresholds in three-dimensional restricted Euler-Poisson equations. Zbl 1434.35105
Lee, Yongki; Liu, Hailiang
2013
Alternating evolution schemes for Hamilton-Jacobi equations. Zbl 1270.65049
Liu, Hailiang; Pollack, Michael; Saran, Haseena
2013
Recovery of normal derivatives from the piecewise $$L^2$$ projection. Zbl 1408.65009
Huang, Yunqing; Liu, Hailiang; Yi, Nianyu
2012
An entropy satisfying conservative method for the Fokker-Planck equation of the finitely extensible nonlinear elastic dumbbell model. Zbl 1422.74097
Liu, Hailiang; Yu, Hui
2012
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit. Zbl 1395.76047
Degond, P.; Liu, H.; Savelief, D.; Vignal, M.-H.
2012
Global well-posedness for the microscopic FENE model with a sharp boundary condition. Zbl 1233.35109
Liu, Hailiang; Shin, Jaemin
2012
The Cauchy-Dirichlet problem for the finitely extensible nonlinear elastic dumbbell model of polymeric fluids. Zbl 1257.35144
Liu, Hailiang; Shin, Jaemin
2012
Alternating evolution schemes for hyperbolic conservation laws. Zbl 1248.65092
Saran, Haseena; Liu, Hailiang
2011
A level set approach for dilute non-collisional fluid-particle flows. Zbl 1391.76348
Liu, Hailiang; Wang, Zhongming; Fox, Rodney O.
2011
The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Zbl 1364.65200
Liu, Hailiang; Yan, Jue
2010
Recovery of high frequency wave fields from phase space-based measurements. Zbl 1221.35031
Liu, Hailiang; Ralston, James
2010
Recovery of high frequency wave fields for the acoustic wave equation. Zbl 1213.35284
Liu, Hailiang; Ralston, James
2010
Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations. Zbl 1218.35175
Liu, Hailiang; Yin, Zhaoyang
2010
Global regularity of the 4D restricted Euler equations. Zbl 1193.37123
Liu, Hailiang; Tadmor, Eitan; Wei, Dongming
2010
The direct discontinuous Galerkin (DDG) methods for diffusion problems. Zbl 1189.65227
Liu, Hailiang; Yan, Jue
2009
Kinetic models for polymers with inertial effects. Zbl 1183.76614
Degond, Pierre; Liu, Hailiang
2009
Critical thresholds in hyperbolic relaxation systems. Zbl 1170.35062
Li, Tong; Liu, Hailiang
2009
Critical thresholds in a relaxation system with resonance of characteristic speeds. Zbl 1171.35428
Li, Tong; Liu, Hailiang
2009
Development of an analytical $$\beta$$-function PDF integration algorithm for simulation of non-premixed turbulent combustion. Zbl 1405.80011
Lien, F. S.; Liu, H.; Chui, E.; McCartney, C. J.
2009
Computational aerodynamics of low Reynolds number plunging, pitching and flexible wings for MAV applications. Zbl 1271.76285
Shyy, W.; Lian, Y.; Tang, J.; Liu, H.; Trizila, P.; Stanford, B.; Bernal, L.; Cesnik, C.; Friedmann, P.; Ifju, P.
2008
On discreteness of the Hopf equation. Zbl 1149.65071
Liu, Hai-liang
2008
Robust control of robotic manipulators based on integral sliding mode. Zbl 1154.93344
Shi, J.; Liu, H.; Bajcinca, N.
2008
Boundary conditions for the microscopic FENE models. Zbl 1147.76003
Liu, Chun; Liu, Hailiang
2008
Superposition of multi-valued solutions in high frequency wave dynamics. Zbl 1203.65141
Liu, Hailiang; Wang, Zhongming
2008
A nonlocal conservation law with nonlinear “radiation” inhomogeneity. Zbl 1158.35063
di Francesco, Marco; Fellner, Klemens; Liu, Hailiang
2008
Global attractor for a nonlinear thermoviscoelastic system in shape memory alloys. Zbl 1173.35386
Qin, Yuming; Liu, Hailiang; Song, Changming
2008
Critical thresholds in a relaxation model for traffic flows. Zbl 1152.35006
Li, Tong; Liu, Hailiang
2008
An alternating evolution approximation to systems of hyperbolic conservation laws. Zbl 1162.35049
Liu, Hailiang
2008
Rigorous derivation of the hydrodynamical equations for rotating superfluids. Zbl 1144.81462
Liu, Hailiang; Sparber, Christof
2008
A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations. Zbl 1168.65056
Liu, Hailiang; Wang, Zhongming
2008
Surfaces in the lightlike cone. Zbl 1108.53011
Liu, H. L.
2007
Computing multi-valued velocity and electric fields for 1D Euler-Poisson equations. Zbl 1146.78012
Liu, Hailiang; Wang, Zhongming
2007
A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations. Zbl 1140.81399
Liu, Hailiang; Wang, Zhongming
2007
Global orientation dynamics for liquid crystalline polymers. Zbl 1183.82101
Liu, Hailiang
2007
The modular multisensory DLR-HIT-Hand. Zbl 1183.70026
Liu, H.; Meusel, P.; Seitz, N.; Willberg, B.; Hirzinger, G.; Jin, M. H.; Liu, Y. W.; Wei, R.; Xie, Z. W.
2007
A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect. Zbl 1092.65083
Liu, Hailiang; Yan, Jue
2006
Wave breaking in a class of nonlocal dispersive wave equations. Zbl 1110.35069
Liu, Hailiang
2006
Multi-valued solution and level set methods in computational high frequency wave propagation. Zbl 1120.65110
Liu, H.; Osher, S.; Tsai, R.
2006
A level set framework for capturing multi-valued solutions of nonlinear first-order equations. Zbl 1109.65080
Liu, Hailiang; Cheng, Li-Tien; Osher, Stanley
2006
Critical thresholds in the semiclassical limit of 2D rotational Schrödinger equations. Zbl 1089.35052
Liu, Hailiang
2006
Sharp interface Cartesian grid method. I: An easily implemented technique for 3D moving boundary computations. Zbl 1154.76359
Marella, S.; Krishnan, S.; Liu, H.; Udaykumar, H. S.
2005
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation. Zbl 1072.65132
Jin, Shi; Liu, Hailiang; Osher, Stanley; Tsai, Yen-Hsi Richard
2005
Sharp interface Cartesian grid method. II: A technique for simulating droplet interactions with surfaces of arbitrary shape. Zbl 1154.76358
Liu, H.; Krishnan, S.; Marella, S.; Udaykumar, H. S.
2005
Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential. Zbl 1092.76007
Liu, Hailiang; Zhang, Hui; Zhang, Pingwen
2005
Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. Zbl 1098.65094
Jin, Shi; Liu, Hailiang; Osher, Stanley; Tsai, Richard
2005
Stability of a traffic flow model with nonconvex relaxation. Zbl 1099.35156
Li, Tong; Liu, Hailiang
2005
Semi-active fuzzy sliding mode control of full vehicle and suspensions. Zbl 1182.70067
Liu, H.; Nonami, K.; Hagiwara, T.
2005
Relaxation dynamics, scaling limits and convergence of relaxation schemes. Zbl 1094.35078
Liu, Hailiang
2005
Molecular dynamics simulations of polyampholytes inside a slit. Zbl 1074.82555
Feng, J.; Liu, H.; Hu, Y.
2005
Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Zbl 1098.76603
Chen, Gui-Qiang; Liu, Hailiang
2004
Rotation prevents finite-time breakdown. Zbl 1098.76643
2004
KdV dynamics in the plasma-sheath transition. Zbl 1060.76137
Liu, Hailiang; Slemrod, M.
2004
Pattern formation, wave propagation and stability in conservation laws with slow diffusion and fast reaction. Zbl 1065.35156
Fan, Haitao; Liu, Hailiang
2004
Formation of $$\delta$$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. Zbl 1038.35035
Chen, Gui-Qiang; Liu, Hailiang
2003
Computational high-frequency wave propagation using the level set method with applications to the semi-classical limit of Schrödinger equations. Zbl 1084.35066
Cheng, Li-Tien; Liu, Hailiang; Osher, Stanley
2003
Critical thresholds in 2D restricted Euler-Poisson equations. Zbl 1073.35187
2003
...and 36 more Documents
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#### Cited by 832 Authors

 65 Liu, Hailiang 20 Yang, Hanchun 16 Shu, Chi-Wang 15 Shen, Chun 14 Huang, Yunqing 13 Sun, Meina 13 Zhang, Yu 12 Cheng, Hongjun 12 Jin, Shi 12 Yi, Nianyu 12 Yin, Gan 10 Wang, Zhongming 9 Shao, Zhiqiang 9 Yu, Xijun 8 Guo, Lihui 8 Huang, Chaobao 8 Yan, Jue 8 Zhang, Pingwen 7 An, Na 7 Lee, Yongki 7 Li, Tong 7 Nedeljkov, Marko 7 Pang, Yicheng 7 Tadmor, Eitan 7 Wang, Jinhuan 7 Xu, Yan 7 Zhang, Yanyan 6 Cheng, Jian 6 Degond, Pierre 6 Fox, Rodney O. 6 Liu, Tiegang 6 Meng, Xiong 6 Osher, Stanley Joel 6 Runborg, Olof 6 Sarrico, Carlos Orlando R. 6 Sheng, Wancheng 6 Wang, Wei 6 Yang, Xu 5 Carrillo de la Plata, José Antonio 5 Liu, Jinjing 5 Mitrovic, Darko 5 Qian, Jianliang 5 Süli, Endre E. 5 Tan, Changhui 5 Wen, Xin 5 Wu, Boying 5 Xing, Yulong 5 Zhang, Rongpei 5 Zhu, Changjiang 4 Baccouch, Mahboub 4 Barrett, John W. 4 Cheng, Yingda 4 Choi, Young-Pil 4 Dai, Limei 4 Gosse, Laurent 4 He, Dongdong 4 Motamed, Mohammad 4 Pan, Kejia 4 Pu, Xueke 4 Tao, Qi 4 Titi, Edriss Saleh 4 Tsai, Yen-Hsi Richard 4 Xia, Yinhua 4 Yin, Peimeng 4 Yong, Wen-An 4 Yuen, Manwai 4 Zhang, Dazhi 4 Zhang, Mengping 4 Zhang, Zhifei 4 Zhang, Zhiyue 4 Zhou, Hong 4 Zhou, Shenggao 4 Zhu, Jiang 3 Cao, Waixiang 3 Chen, Zheng 3 Cheng, Bin 3 Cheng, Yao 3 Cockburn, Bernardo 3 Dehghan Takht Fooladi, Mehdi 3 Du, Qiang 3 Gottlieb, Sigal 3 Huang, Hongying 3 Karlsen, Kenneth Hvistendahl 3 Klar, Axel 3 Kwon, Bongsuk 3 LeFloch, Philippe Gerard 3 Li, Shiwei 3 Li, Xianggui 3 Loula, Abimael Fernando Dourado 3 Lukáčová-Medvid’ová, Mária 3 Luo, Hong 3 Markowich, Peter Alexander 3 Mei, Ming 3 Mizerová, Hana 3 Natalini, Roberto 3 Nishikawa, Hiroaki 3 Qu, Aifang 3 Rohde, Christian 3 Rozanova, Olga S. 3 Ruan, Lizhi ...and 732 more Authors
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#### Cited in 140 Serials

 78 Journal of Computational Physics 49 Journal of Scientific Computing 46 Journal of Differential Equations 33 Journal of Mathematical Analysis and Applications 21 Journal of Mathematical Physics 20 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 19 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 16 Mathematics of Computation 16 Journal of Computational and Applied Mathematics 11 ZAMP. Zeitschrift für angewandte Mathematik und Physik 11 Physica D 11 Applied Numerical Mathematics 11 Journal of Hyperbolic Differential Equations 10 Archive for Rational Mechanics and Analysis 10 Computers & Mathematics with Applications 10 Communications on Pure and Applied Analysis 9 Computers and Fluids 9 Applied Mathematics and Computation 9 Numerische Mathematik 9 SIAM Journal on Numerical Analysis 9 Applied Mathematics Letters 8 Mathematical Methods in the Applied Sciences 8 SIAM Journal on Mathematical Analysis 8 Nonlinear Analysis. Real World Applications 8 Advances in Mathematical Physics 7 Acta Applicandae Mathematicae 7 SIAM Journal on Scientific Computing 6 Wave Motion 6 Chinese Annals of Mathematics. Series B 6 Numerical Algorithms 5 Applicable Analysis 5 Communications in Mathematical Physics 5 Computer Methods in Applied Mechanics and Engineering 5 Communications on Pure and Applied Mathematics 5 Abstract and Applied Analysis 5 Science China. Mathematics 4 SIAM Journal on Applied Mathematics 4 Physics of Fluids 4 Discrete and Continuous Dynamical Systems 4 Nonlinear Dynamics 4 European Journal of Mechanics. B. Fluids 4 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 4 Boundary Value Problems 3 Nonlinearity 3 BIT 3 Monatshefte für Mathematik 3 Quarterly of Applied Mathematics 3 Numerical Methods for Partial Differential Equations 3 Applied Mathematical Modelling 3 Communications in Partial Differential Equations 3 International Journal of Computer Mathematics 3 Computational and Applied Mathematics 3 Advances in Computational Mathematics 3 Mathematical Problems in Engineering 3 Mathematics and Mechanics of Solids 3 Journal of Nonlinear Mathematical Physics 3 Multiscale Modeling & Simulation 3 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 3 Kinetic and Related Models 3 Journal of Applied Analysis and Computation 2 Journal of Statistical Physics 2 Bulletin of Mathematical Biology 2 Advances in Mathematics 2 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 2 Computational Mathematics and Mathematical Physics 2 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2 Applied and Computational Harmonic Analysis 2 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 2 Journal of Mathematical Fluid Mechanics 2 Communications in Nonlinear Science and Numerical Simulation 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 2 Acta Numerica 2 Research in the Mathematical Sciences 2 Communications on Applied Mathematics and Computation 1 International Journal of General Systems 1 Indian Journal of Pure & Applied Mathematics 1 Journal of Engineering Mathematics 1 Journal of Mathematical Biology 1 Moscow University Mathematics Bulletin 1 Physics Letters. A 1 Rocky Mountain Journal of Mathematics 1 Theoretical and Mathematical Physics 1 Zhurnal Vychislitel’noĭ Matematiki i Matematicheskoĭ Fiziki 1 Journal of Geometry and Physics 1 Duke Mathematical Journal 1 International Journal for Numerical Methods in Engineering 1 Inventiones Mathematicae 1 Journal of Functional Analysis 1 Journal of the Korean Mathematical Society 1 Mathematics and Computers in Simulation 1 Pacific Journal of Mathematics 1 Proceedings of the American Mathematical Society 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Ricerche di Matematica 1 Transactions of the American Mathematical Society 1 Applied Mathematics and Mechanics. (English Edition) 1 Acta Mathematicae Applicatae Sinica. English Series 1 Mathematical and Computer Modelling 1 Asymptotic Analysis 1 Forum Mathematicum ...and 40 more Serials
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#### Cited in 32 Fields

 524 Partial differential equations (35-XX) 287 Numerical analysis (65-XX) 243 Fluid mechanics (76-XX) 68 Statistical mechanics, structure of matter (82-XX) 31 Biology and other natural sciences (92-XX) 25 Optics, electromagnetic theory (78-XX) 22 Quantum theory (81-XX) 20 Dynamical systems and ergodic theory (37-XX) 17 Mechanics of deformable solids (74-XX) 14 Operations research, mathematical programming (90-XX) 13 Functional analysis (46-XX) 12 Ordinary differential equations (34-XX) 10 Relativity and gravitational theory (83-XX) 8 Astronomy and astrophysics (85-XX) 8 Geophysics (86-XX) 7 Calculus of variations and optimal control; optimization (49-XX) 6 Probability theory and stochastic processes (60-XX) 5 Global analysis, analysis on manifolds (58-XX) 4 Real functions (26-XX) 4 Approximations and expansions (41-XX) 4 Integral equations (45-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Mechanics of particles and systems (70-XX) 1 Mathematical logic and foundations (03-XX) 1 Topological groups, Lie groups (22-XX) 1 Measure and integration (28-XX) 1 Difference and functional equations (39-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Operator theory (47-XX) 1 Computer science (68-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Systems theory; control (93-XX)