臺灣博碩士論文加值系統

本論文在研究網格調整的想法過程，並探討一些依據平均分配原則 ( equidistribution principle ) 所造出的移動網格偏微分方程式 ( moving mesh partial differential equations )。同時，利用移動網格代數方程式 ( moving mesh differential algebraic equations ) 所衍生的方法，我們也作了理論及計算方面的分析。最後，我們利用移動網格法實際做一個數值實驗。

In this thesis, we survey the idea of grid adaption and some moving
mesh partial differential equations that are related to the
equidistribution principle. We study some theoretical and computational
aspects of the methods under the moving mesh differential algebraic
equations (DAEs) framework. Finally, we give a numerical experiment
solved by moving mesh method.

1. Introduction
2. Grid Adaption
2.1 measure the "error"
2.2 monitor function
3. Mesh selection problem
3.1 equidistribution principle
3.2 transformation method
4. Moving mesh methods
4.1 moving mesh PDEs
4.1.1 construct MMPDEs
4.1.2 theoretical analysis of MMPDEs
4.2 moving mesh DAEs
4.2.1 analysis of convergence
5. Numerical Experiment
5.1 available solvers
5.2 Medical Akzo Nobel problem
5.3 numerical results and conclusion

[1]S. Adjerid and E. Flaherty. A moving finite element method with error estimate and refinement for one-dimensional time dependent
partial differential equations, SIAM J. Numer. Anal., 1986.
[2]S. Adjerid and E. Flaherty. A moving-mesh finite element method with local refinement for parabolic partial differential equations.
Comput. Meth. Appl. Mech. Engrg., 1986.
[3]D.A. Anderson. Adaptive mesh schemes based on grid speeds., AIAA Paper., 1983.
[4]D.C. Arney and J.E. Flaherty. A two-dimensional mesh moving technique for time-dependent partial differential equations. J. Comput. Phys., 1986.
[5]J. Baumgarte. Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Engrg., 1972.
[6]J. Cash. Mebdfade.
Available at http://www.ma.ic.ac.uk/jcash/IVP\_ software/finaldae/readme.html. 1998.
[7]C. deBoor. Good approximation by splines with variable knots. ii. Conference on the Numerical Solution of Differential Equations,
Lecture Notes in Mathematics, 1973.
[8]D.S. Dodson. Optimal order approximation by polynomial spline functions. ph.D. Thesis, Purdue Univ., Lafayette,IN, 1972.
[9]E.A. Dorfi and L.O'c. Drury. Simple adaptive grids for 1-d initial value problem. J.Comput.Phys., 1987.
[10]E. Hairer and G. Wanner. Radau5.
Available at ftp://ftp.unige.ch/pub/doc/math/stiff/radau5.f, 1996.
[11]E. Hairer and G. Wanner. Radau.
Available at ftp://ftp.unige.ch/pub/doc/math/stiff/radau.f, 1998.
[12]R.G.Hindman and J. Spencer. A new approach to truly adaptive grid generation. AIAA Paper 83-0450, 1983.
[13]J.M. Hyman and M.J. Naughton. Static rezone methods for tensor-product grids. in Proceedings of SIAM-AMS Conference on Large Scale Computations in Fluid Mechanics,SIAM,Philadelphia, 1984.
[14]R.Ludwig J.E. Flaherty, J.M. Coyle and S.F. Davis. Adaptive finite element methods for parabolic partial differential equations.
Society for Industrial and Applied Mathematics, Philadephia, PA, 1983.
[15]W.M. Lioen J.J.B. de Swart and W.A. van der Veen. Pside.
Available at http://www.cwi.nl/cwi/projects/PSIDE/, 1998.
[16]K. Miller and R.N. Miller. Moving finite elememts i. SIAM J. Numer. Anal., 1981.
[17]L.R. Petzold. DASSL: A Differential/Algebraic System Solver.
Available at http://www.netlib.org/ode/ddassl.f, 1991.
[18]Y.Ren. Theory and computation of moving mesh methods for solving time-dependent partial differential equations. phD. thesis,
Department of Mathematics and Statistics, Simon Fraser University, Burnday,Canada, 1992.
[19]Y. Ren and R.D. Russell. Moving mesh techniques based upon equidistribution, and their stability. SIAM J. Statist. Comput., 1992.
[20]R.D. Russell and J. Christiansen. Adaptive Mesh Selection Strategies for Solving Boundary Value Problems. SIAM J. Sci. Comput., 1998.
[21]Shengtai Li, Linda Petzold, and Yuhe Ren. Stability of Moving Mesh Systems of Partial Differential Equations. SIAM J. Sci.
Comput., 1998.
[22]A.B.White. On selection of equidistributing meshes for two-point boundary-value problems. SIAM J. Numer. Anal., 1979.