Research Interests

My research focuses primarily on the analysis and implementation of high resolution, non-oscillatory numerical schemes for hyperbolic system of conservation laws and related problems. In particular, I am interested in the computational and mathematical challenges posed by the equations of ideal Magnetohydrodynamics (MHD), a hyperbolic model which describes the dynamics of electrically conducting fluids in the presence of a magnetic field.

Other topics of interest include: numerical schemes for PDEs in plasma physics, shallow water flows.

For more information about my research interests, see this research statement (.pdf)

Publications

• Non-oscillatory Central Schemes for 3D Hyperbolic Conservation Laws (with Xin Qian), “Hyperbolic Partial Differential Equations, Theory, Numerics and Applications”, Proceedings of the 12th international conference held at the Univedrsity of Maryland, To appear
  
  
• A Numerical Study of Magnetic Reconnection: A Central Scheme for Hall MHD (X. Qian, A. Bhattacharjee, & H. Yang ), “Hyperbolic Partial Differential Equations, Theory, Numerics and Applications”, Proceedings of the 12th international conference held at the Univedrsity of Maryland, To appear
  
  
• A central Scheme or Shallow Water Flows along Channels with Irregular Geometry. (with S. Karni), M2AN, 43 (2): 333 - 351 (2009).
• Non-oscillatory central schemes for one- and two-dimensional MHD equations. II: high-order semi-discrete schemes. (with E. Tadmor) SIAM J. of Scient. Comput., 28(2): 533-560, 2006.
• A central differencing simulation of the Orszag-Tang vortex system. (with E. Tadmor) 4th Triennial Special Issue of the IEEE Transactions on Plasma Science "Images in Plasma Science" 33 (2) (2005) 470-471.
• Non-oscillatory central schemes for one- and two-dimensional MHD equations. I. (with E. Tadmor & C-c. Wu) J. Comput. Phys., 201(1): 261-285, 2004.

Selected Lectures

• High-order central schemes for hyperbolic systems of conservation laws
• MHD, the div(B) constraint, and central schemes

Examples and Simulations

Part of my research focuses in the development of software tools for scientific computing. In particular, I am currently working on developing a collection of libraries that implement different versions of high-order non-oscillatory central schemes for hyperbolic conservation laws. Although we are still testing and improving this code, we hope it will be available to the public in the near future. In the mean time, you can find examples of hyperbolic conservation laws that I have solved using these libraries. Click in the links below to see a description of the equations and some animations.

• Scalar conservation laws in one space dimension:
  • Burgers' equation
  • Buckley-Leverett equation
• Systems of conservation laws in one space dimension:
  • Euler's equations of gas dynamics
  • Ideal MHD equations
• Systems of conservation laws in two space dimension:
  • Euler's equations of gas dynamics
  • Ideal MHD equations
• Systems of conservation laws in three space dimension:
  • Euler's equations of gas dynamics