Principal Investigator: Jun Zhang
Co-Principal Investigator: Craig C. Douglas
Current Postdoctoral Research Associate: Samir Karaa
Former Postdoctoral Research Associate: Lixin Ge (07/01/2000 - 12/30/2000), Haiwei Sun (02/01/2001 - 07/30/2002)

Abstract:

This project will design efficient geometric multigrid methods using intermediate grids to solve convection diffusion problems discretized by high order compact schemes. The use of intermediate grids aims at reducing discrepancy between the solutions obtained on different grids. Symbolic computation packages will be used to derive a fourth order finite difference scheme on the intermediate grids. Special relaxation schemes and intergrid transfer operators will be developed. The nonstandard multigrid method with high order discretization schemes will be used in the numerical simulation of laminar diffusion flames. The use of intermediate grids is expected to alleviate the problem haunting existing multigrid methods in the flame simulation, in which the converged coarse grid correction is not in the convergence domain of the fine grid Newton iteration.

This project requires both symbolic and numerical computation techniques. The successful development of a useful symbolic computation procedure will greatly promote the awareness and use of symbolic computation packages in numerical computation community. The results of this research project will make important contribution to the understanding of geometric multigrid solution of convection diffusion problems, and of the applications of high order compact schemes to realistic flow simulations. Efficient solution of such problems is central to many numerical simulations in computational fluid dynamics. The fast laminar diffusion flame code is useful in combustion and environment protection. Certain U.S. industries related to commercial burners, pollutant tracking, car and airplane manufacturing, combustion, can benefit from this research.

Technical Reports and Computer Software:

1. A family of fourth order difference schemes on rotated grid for two dimensional convection-diffusion equation, by Jun Zhang, Jules Kouatchou, and Lixin Ge. (December, 2000).
2. Multigrid method and fourth order compact scheme for 2D Poisson equation with unequal meshsize discretization, by Jun Zhang. (March, 2001).
3. Maple symbolic computation codes for computing fourth order compact schemes,. (April 2001).
4. Truncation error and oscillation property of the combined compact difference scheme, by Jun Zhang and Jennifer J. Zhao, (October 2001).
5. High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems, by Jun Zhang, Haiwei Sun, and Jennifer J. Zhao, (October 2001).
6. A high order compact boundary value method for solving one dimensional heat equations, by Haiwei Sun and Jun Zhang. (January, 2002).
7. A three level finite difference scheme for solving the Pennes' bioheat transfer in a triple-layered skin structure, by Weizhong Dai, Raja Nassar, and Jun Zhang. (May, 2002).
8. A high order finite difference discretization strategy based on extrapolation for convection diffusion equations, by Haiwei Sun, and Jun Zhang. (June, 2002).
9. The ASEI scheme for solving the dispersive equation with diffusion, by Shaohong Zhu, and Jennifer J. Zhao, and Jun Zhang. (July, 2002).
10. A two level finite difference scheme for one dimensional Pennes' bioheat equation, by Jennifer J. Zhao, Jun Zhang, Ning Kang, and Fuqian Yang. (September, 2002).
11. Preconditioned multigrid simulation of an axisymmetric laminar diffusion flame, by Samir Karaa, Jun Zhang, and Craig C. Douglas. (October, 2002).
12. A numerical study of a 3D bioheat transfer with different spatial heating, by Samir Karaa, Jun Zhang, and Fuqian Yang. (April, 2003).
13. High order ADI method for solving unsteady convection-diffusion problems, by Samir Karaa, and Jun Zhang. (June, 2003).
14. Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue by Wensheng Shen, Jun Zhang, and Fuqian Yang (January 27, 2004).
15. Three-dimensional model on thermal response of skin subject to laser heating by Wensheng Shen, Jun Zhang, and Fuqian Yang (July 26, 2004).

Conference and Seminar Presentations and Posters, and Special Session Organizations:

1. Minisymposium: High Order Compact Discretization Schemes and Applications, organized by Jun Zhang, First SIAM on Computational Science and Engineering, September 21 - 24, 2000, Washington, DC, USA.
2. High order compact scheme and iterative methods for 3D boundary layer problems, presented by Jun Zhang, First SIAM on Computational Science and Engineering, September 21 - 24, 2000, Washington, DC, USA.
3. Relation between cyclic reduction and finite difference schemes, presented by Jun Zhang, Colloquium, Department of Mathematics, George Washington University, September 24, 2000, Washington, DC, USA.
4. Multigrid method and high order compact scheme for solving boundary layer problems on nonuniform grids, presented by Jun Zhang, Tenth Copper Mountain Conference on Multigrid Methods, April 1 - 6, 2001, Copper Mountain, CO, USA.
5. Recursive-based PCG methods for Toeplitz systems, presented by Haiwei Sun, Midwest Numerical Analysis and Scientific Computing Day, May 12, 2001, Urbana-Champaign, IL, USA.
6. Minisymposium: Advanced Computational and Modeling Techniques in Computer Simulations of Physical and Engineering Processes, organized by Jennifer J. Zhao and Jun Zhang, 2001 SIAM Annual Meeting, July 9 - 13, 2001, San Diego, CA, USA.
7. Finite difference approximations to microscale heat transport equations, presented by Jennifer J. Zhao. 2001 SIAM Annual Meeting, July 9 - 13, 2001, San Diego, CA, USA.
8. High order compact scheme and multigrid local refinement for convection diffusion problems, presented by Jun Zhang, 2001 Fall AMS Southeastern Sectional Meeting, October 5 - 6, 2001, Chattanooga, TN, USA.
9. High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems, presented by Jun Zhang, 2002 SIAM 50th Anniversary and Annual Meeting, July 8 - 12, 2002, Philadelphia, PA, USA.
10. High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems, presented by Jennifer J. Zhao, International Conference on Nonlinear Partial Differential Equations, August 29 - September 2, 2002, Hongkong, China.
11. Multilevel multigrid high accuracy solution of convection diffusion equation with local refinement, (Jun Zhang, Haiwei Sun, Jennifer J. Zhao), presented (by Jun Zhang) at the Sixth IMACS International Symposium on Iterative Methods in Scientific Computing, Denver, CO, March 27 - 30, 2003.
12. Preconditioned multigrid simulation of an axisymmetric laminar diffusion flame, (Samir Karaa, Jun Zhang, Craig C. Douglas), presented (by Samir Karaa) at the conference of ETNA: Following the Flows of Numerical Analysis, Kent, OH, May 29-31, 2003.
13. A sixth order finite difference scheme for the convection diffusion equation, (Jun Zhang, Haiwei Sun), presented (by Jun Zhang) at the Second MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 17-20, 2003.
14. Modeling and simulation of heat transfer and biomechanics in soft tissue, (Wensheng Shen, Jun Zhang, Fuqian Yang), presented (by Wensheng Shen) at the SIAM Conference on Life Science, Portland, Oregon, July 10-14, 2004.

This page is supported by the U.S. National Science Foundation. However, any opinions, findings, and conclusions or recommendations expressed in this documents are those of the author and do not necessarily reflect the views of the U.S. National Science Foundation.

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