Research interests: Murli M. Gupta
 

My research interests lie primarily in the area of numerical analysis with particular emphasis on the high accuracy solutions of partial differential equations of elliptic type. My recent work has focused on the development of multigrid techniques for the solution of the convection-diffusion equations and the biharmonic equation. Both of these problems are special linearized cases derived from the Navier-Stokes equations which represent most of the fluid flow phenomena. In recent years, I have developed a number of high accuracy, compact difference schemes for solving these problems; these difference schemes are currently being used in conjunction with the multigrid and other techniques to obtain very high accuracy solutions with very low computational costs. The compact difference schemes themselves can also be derived using the symbolic algebra packages such as Maple or Mathematica. I am also interested in the area of numerical linear algebra, in particular on the acceleration of convergence of iterations both converging or diverging. In addition, I continue to examine the role of technology in the instruction of mathematics at the undergraduate level; I have introduced graphing calculators such as HP48 in the teaching of calculus and have introduced Matlab in the teaching of elementary matrix algebra at The George Washington University.

Selected Publications (For a detailed list, contact Professor Gupta at mmg@gwu.edu)

Research Publications

  • Nature of Viscous Flows near Sharp Corners, (with R. Manohar and B. Noble), Computers and Fluids, Vol. 8 (1981), pp. 379-388.
  • A Fourth-Order Poisson Solver, Journal of Computational Physics, Vol. 55 (1984), pp. 166-172.
  • A Single Cell High Order Difference Scheme for the Convection-Diffusion Equation with Variable Coefficients, (with R. Manohar and J.W. Stephenson), International Journal for Numerical Methods in Fluids, Vol. 4 (1984), pp. 641-651.
  • A Spectrum Enveloping Technique for Iterative Solution of Central Difference Approximations of Convection Diffusion Equations, SIAM Journal on Algebraic and Discrete Methods, Vol. 7 (1986), pp. 513-526.
  • High Accuracy Solutions of Incompressible Navier-Stokes Equations, Journal of Computational Physics, Vol. 93 (1991), pp. 343-359.
  • Comparison of Second and Fourth Order Discretizations for Multigrid Poisson Solvers, (with J. Kouatchou and J. Zhang), Journal of Computational Physics, Vol. 132 (1997), pp. 226-232.
  • A Compact Multigrid Solver for Convection-Diffusion Equations, (with J. Kouatchou and J. Zhang), Journal of Computational Physics, Vol. 132 (1997), pp. 123-129.
  • Iterative Methods for Fixed Point Problems on High Performance Computing Environment, (with I. Altas), Complex Systems: from Local Interactions to Global Phenomena, (R. Stocker et al , Eds.), IOS Press, Amsterdam, pp. 122-128. Also available online.
  • Multigrid Solution of Automatically Generated High Order Discretizations for the Biharmonic Equation, (with I. Altas, J. Dym and R. Manohar), SIAM Journal on Scientific Computing,Vol. 19 (Sept 1998), pp. 1575-1585. Available online at <http://www.siam.org/journals/sisc/19-5/29697.html>.
  • Symbolic Derivation of finite difference approximations for three- dimensional Poisson equation, (with J. Kouatchou), Numerical Methods for Partial Differential Equations, Vol. 14 (1998), pp. 593-606.

 

  • High accuracy multigrid solution of the 3D convection diffusion equation, (with J. Zhang), Technical Report 278-98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998. Applied Mathematics and Computation, Vol. 113 (2000), pp. 249-274.

 

  • Fourth order compact difference scheme for 3D convection diffusion equation with boundary layers on nonuniform grids, (with  Jun Zhang and Lixin Ge), Neural, Parallel & Scientific Computations, Volume 8 (2000), pp. 373- 392.

 

  • High Accuracy Solution of Two and Three Dimensional Biharmonic Equations, (with I. Altas and J. Erhel), Proceedings of 2nd International Seminar on Numerical Analysis in Engineering (NAE 2001), Batam Island, Indonesia  (March 2001), pp. 5/17 - 5/23 (2001).

 

  • High Accuracy Solution of Three- Dimensional Biharmonic Equations, (with I. Altas and J. Erhel), Numerical Algorithms, Vol. 29  (2002), pp. 1-19.
  • A Stream-function/Velocity Method for High Accuracy Solution of Navier- Stokes  Equations, Book of Abstracts of ICIAM 2003, 5th  International Congress on Industrial and Applied Mathematics, Sydney, Australia  (July 2003), p. 59.

Mathematics Education Articles

  • Graphing Calculators in the Calculus Classroom, SIAM News, Volume 26 (1993), pp. 18, 21.
  • Teaching Linear Algebra with Matlab, Innovative Use of Technology for Teaching and Research in Mathematics, Proc. First Asian Conf. Math., Assoc. Math. Educators, Singapore (1995), pp. 128-134.
  • Five years of Summer Programs for Women in Mathematics at The George Washington University, Proceedings of the Conference on Summer Undergraduate Mathematics Research Programs, September- October  1999, (J. Gallian, Ed.), American Mathematical Society (2000), pp. 237- 244.