A personal view of mathematical research:

• IMS Public Lecture The Mathematics of Scientific Computation
  
• Velocity averaging, kinetic formulations and regularizing effects in conservation laws and related PDEs
  
• On a multiscale representation of images as hierarchy of edges
  
• Processing discontinuous spectral data
  
• Critical thresholds and Eulerian dynamics
  
• Burgers-type equations and vanishing hyper-viscosity
  
• Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity
  
• On a new scale of borderline regularity spaces for Euler equations
  
• Detection, reconstruction and approximate evolution of piecewise smooth solutions [ICM 2002 Lecture]

Selected Publications    Other lists of publications: [by chronological order] [by subject classification] [significant publications]

• Heterophilious dynamics enhances consensus (with S. Motsch), preprint .
  
• A review of numerical methods for nonlinear partial differential equations, Bull. AMS 49(4) (2012) 507-554.
  
• ENO reconstruction and ENO interpolation are stable (with U. Fjordholm & S. Mishra), FoCM 13(2) (2012) 139-159.
  
• A new model for self-organized dynamics and its flocking behavior (with S. Motsch), J. Stat. Physics 144(5) (2011) 923-947.
  
• Hierarchical construction of bounded solutions of div U=F in critical regularity spaces (with C. Tan) ``Nonlinear Partial Differential Equations'', Proc. 2010 Abel Symposium (H. Holden & K. Karlsen eds.), Abel Symposia 7, Springer (2011), pp. 255-269.
  
• Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography (with U. Fjordholm & S. Mishra)
  J. Computational Physics 230 (2011), 5587-5609.
  
• Integro-differential equations based on (BV, L¹) image decomposition (with P. Athavale) SIAM J. Imaging Sciences 4(1) (2011) 300-312.
  
• Global regularity of the 4D Restricted Euler Equations (with H. Liu & D. Wei) Physica D 239 (2010) 1225-1231.
  
• Constraint preserving schemes using potential-based fluxes (with S. Mishra)
  I. Multidimensional transport equations Comm. Comput. Phys. 9(3) (2010) 688-710.
  II. Genuinely multi-dimensional systems of conservation laws. SINUM 49(3) (2011) 1023-1045.
  III. Genuinely multi-dimensional schemes for the MHD equations M2AN 46 (2012) 661-680.
  
• Hyperbolic Problems: Theory, Numerics, Applications Proceedings of HYP2008, Univ. of Maryland, College Park, June 9-13, 2008
  AMS Proc. Symp. Appl. Math., 2009 (with J.-G. Liu & A. Tzavaras) Vol. 67.1: Plenary & Invited Talks; Vol. 67.2: Contributed Talks.
  
• Energy-preserving and stable approximations for the two-dimensional shallow water equations (with W. Zhong) in "Mathematics and Computation - A Contemporary View", Proc. of the Third Abel Symposium (H. Munthe-Kaas & B. Owren eds.), Springer 2008, pp. 67-94.
  
• From particle to kinetic and hydrodynamic descriptions of flocking (with S.-Y. Ha) Kinetic and Related Models 1(3) (2008) 415-435.
  
• L² Stability Analysis of the Central Discontinuous Galerkin Method and a Comparison between the Central and Regular Discontinuous Galerkin Methods (with Y.-J. Liu, C.-W. Shu  & M. Zhang) M2AN 42 (2008) 593-607 [highlighted paper].
• Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation (with S. Nezzar & L. Vese) Comm. Math. Sciences 6(2) (2008) 281-307.
  
• Recovery of edges from spectral data with noise---a new perspective (with S. Engelberg) SINUM 46(5) (2008), 2620-2635.
  
• Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations (with B. Cheng)
  SIAM J. Math. Anal. 39(5) (2008) 1668-1685.
  
• Filters, mollifiers and the computation of the Gibbs phenomenon Acta Numerica 16 (2007) 305-378.
  
• Velocity averaging, kinetic formulations and regularizing effects in quasilinear PDEs (with T. Tao) CPAM 60 (2007), 1488-1521.
  

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• Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems
  Acta Numerica 12 (2003), 451-512.
  
• Hyperbolic Problems: Theory, Numerics, Applications Proceedings of the Ninth International Conference on Hyperbolic Problems held in CalTech, Pasadena, March 25-29, 2002 (with T. Hou), Springer-Verlag, ISBN: 3-540-44333-9, 2003. Table of content
  
• High resolution methods for time dependent problems with piecewise smooth solutions "International Congress of Mathematicians", Proceedings of the ICM02 Beijing 2002 (Li Tatsien, ed.), Vol. III: Invited lectures, Higher Education Press (2002) 747-757.
  
• On a new scale of regularity spaces with applications to Euler's equations Nonlinearity 14 (2001) 513-532.
  
• Strong stability-preserving high-order time discretization methods (with S. Gottlieb & C.-W. Shu) SIAM Review 43 (2001) 89-112.
  
• New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations (with A. Kurganov)
  JCP, 160 (2000) 241-282.
  
• Approximate solutions of the incompressible Euler equations with no concentrations (with M. Lopes Filho & H. J. Nussenzveig) Annal. De L'institut Henri Poincare (c) Non Linear Analysis, 17 (2000) 371-412.
  
• Approximate solution of nonlinear conservation laws and related equations "Recent Advances in Partial Differential Equations and Applications" 1996 Venice Conference R. Spigler and S. Venakides eds.), AMS Proc. Symp. Appl. Math. 54 (1998) 325-368.
  
• A kinetic formulation of multidimensional scalar conservation laws and related equations (with P.-L. Lions & B. Perthame)
  J. Amer. Math. Soc. 7 (1994) 169-191.
  
• On the stability of the unsmoothed Fourier method for hyperbolic equations (with J. Goodman & T. Hou) Numer. Math. 67(1) (1994), 93-129.
  
• Local error estimates for discontinuous solutions of nonlinear hyperbolic equations SINUM 28 (1991) 891-906.
  
• A kinetic equation with kinetic entropy functions for scalar conservation Laws (with B. Perthame) Comm. Math. Phys. 136 (1991) 501-517.
  
• Non-oscillatory central differencing for hyperbolic conservation laws (with H. Nessyahu) JCP 87 (1990) 408-463.
  
• An O(N²) method for computing the eigensystem of N x N symmetric tridiagonal matrices by the divide and conquer approach (with D. Gill) SIAM J. Scientific Computing 11 (1990) 161-173.
  
• Convergence of spectral methods for nonlinear conservation laws SINUM 26 (1989) 30-44.
  
• Numerical viscosity and the entropy condition for conservative difference schemes Math. Comp. 43 (1984) 369-381
  
• The equivalence of L²-stability, the resolvent condition and strict H-stability LAA 41 (1981) 151-159.
  
• Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II (with M. Goldberg)
  Math. Comp. 36 (1981) 603-626.
  

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