Current Research Projects (2007-2008)

Information for current students or postdoc applicants: 

• Together with Jean-Luc Guermond we have developed a convergence theory for numerical methods on triangulations based on $L_1$ minimization for  2-D Hamilton-Jacobi equation with convex hamiltonian. To the best of my knowledge, this is the only result for convergence of second or higher order schemes for such nonlinear equations. This project is supported by the NSF grant DMS-0510650.
  
• Together with  O. Trifonov, we have proved convergence of the Nessyahu-Tadmor scheme based on the minmod limiter to the entropy solution of a scalar conservation law with strictly convex flux. Up to now, no convergence was proven for that scheme and any other central second order scheme. The future goal is to prove error estimates for other second or higher order schemes and  derive error estimates.
  
• Together with A. Kurganov and G. Petrova, we wrote a paper about the construction and analysis of a Minmod-type schemes based on adaptive limiters.  I continued this type of research with Ivan Christov (M.S. 2006) and our joint work New nonoscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws,  will appear in J. Comput. Phys. in 2008.

Linear transport and advection reaction equations

• I am working on existence, stability, regularity and uniqueness of continuous solutions of linear transport equations with discontinuous velocity. The difficulty comes from the fact that the weak solutions are, in general, not unique and one needs to define a uniqueness criteria. There are many open problems, especially in more than one space dimension. I have many results in two space dimension and I am preparing a paper that treats that case.
• Together with Ivan Christov and Peter Popov, we are developing classes of explicit numerical methods for approximations of  nonlinear conservation laws and Hamilton-Jacobi equations. The main new element is that the methods are high-order and based on unstructured triangulations. There is some interest in this work from Exxon-Mobil and Ivan Christov is going to Houston to work for them in the Summer of 2008.
  

• PhD students -- there are many interesting theoretical and numerical problems in the above fields and we (together with Jean-Luc Guermond) have the funding to support graduate students;
• Postdocs -- together with Jean-Luc Guermond we have supported three postdocs and if the funding trends continue will have the money to support more. Send me an email if you are interested.