時空守恆法(CESE method)為一能提供高解析度、有效減低數值耗散與震盪之嶄新數值方法，有別於傳統數值方法，時空守恆法將空間一階導數視為變數，並將時間及空間同等對待，強制物理通量於時間及空間上滿足局部和全域內均保持守恆，此等特點使得時空守恆法成為模擬具有不連續分布之物理現象問題理想的數值方法，已成功的應用於震波捕捉、象變化研究、熱波研究及淺水波模擬等。
電泳技術為生醫研究上所不可或缺的實驗方法，隨著電泳分離技術與應用的拓展，其相關模擬方法也日益發展，用以輔佐實驗進行及設計。本篇論文將時空守恆法良好解析之特性延伸至電泳分離、堆疊、聚焦技術之研究，利用時空守恆法模擬區間電泳法(ZE)、等速電泳法(ITP)及等電點聚焦電泳(IEF)技術，和傳統數值方法進行一系列比較，結果證實時空守恆法確實能提供高計算效率及高可靠度之數值解。此外，研究中亦推導含有面積效應之一維質量傳輸模型並加以驗證，此方法能夠將具有幾何效應的電泳堆疊現象之二維模擬簡化，於短時間內提供分離、傳輸現象之可靠數值解，因此大大增加此問題的計算效益。
除上述之應用方面研究，吾人亦發展非均勻網格時空守恆法以結合適應性網格重新分布技術應用於電擁分離、聚焦問題。由於時空守恆法受庫倫數之限制，並且於過小庫倫數(CFL number<0.1)狀態下會產生數值耗散現象，因此再輔以庫倫數非敏感性技術(CFL number insensitive scheme)更能增進計算準確度。經由驗證，適應性網格時空守恆法於等速電泳法及等電點聚焦電泳法顯示具有高計算效率及準確度。此研究，不但拓展時空守恆法的應用性，更提供一個有效準確之電泳模擬工具。

The space-time conservation element and solution element (CESE) method provides a powerful numerical tool for solving a diverse range of problems in the continuum mechanics domain. The CESE method suppresses the effects of numerical dissipation and oscillation, and is therefore more accurate and robust than traditional numerical schemes such as finite difference method. In the CESE method, both the independent variables and their derivatives are treated as unknowns and are solved simultaneously. Furthermore, the CESE scheme treats the space and time domains in a unified fashion and enforces both local and global flux conservation in the space-time domain. As a result, the CESE method is an ideal solver for wave problems characterized by discontinuous phenomena or sharp gradients, such as shock waves, phase change phenomena, thermal waves, and so on.
Electrophoresis plays a key role in biomedical science in fractionating mixtures of ionic solutes for analytical and preparative applications. Due to the miniature scale of electrophoretic systems, electrophoresis phenomena are generally investigated using some form of numerical simulation technique prior to experimental investigations. In this dissertation, the CESE method is used to investigate various electrophoresis separation phenomena, including zone electrophoresis (ZE), isotachophoresis (ITP) and isoelectric focusing (IEF). The solutions are compared with those obtained from traditional numerical schemes such as finite difference scheme and finite volume method. It is shown that the CESE method is not only more computationally efficient than traditional numerical schemes, but also yields more accurate results. In addition, a 1-D mass transport model is proposed to describe the electrophoresis transport behavior within a microchannel with a variable cross-sectional area. The validity of the proposed model is confirmed by comparing the solutions with those obtained from the 2-D finite volume method (FVM). It is shown that the modified mass transport model provides reliable results with a low computational cost for electrophoresis transport problems within simplified-geometry channels.
Finally, this dissertation proposes an adaptive mesh redistribution (AMR) CESE scheme for the solution of the species transport equation in electrophoresis preconcentration and separation problems. To prevent numerical dissipation at very small values of the CFL number (i.e. <0.1), the spatial differential component within the CESE formulation is treated using a CFL number insensitive scheme. The results obtained for various ITP and IEF problems show that the AMR-CESE scheme improves the solution quality relative to that achieved using a uniform fixed-mesh solver whilst incurring no more than a moderate increase in the computational cost. This dissertation not only extends the application of the conventional CESE method, but also develops a powerful simulation method for general electrophoresis problems.

Beckett, G. and Mackenzie J. A., “Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem”, Appl. Numer. Math., Vol. 35, pp. 87－109, 2000
Bercovici M., Lele S. K. and Santiago J. G., “Open source simulation tool for electrophoretic stacking, focusing, and separation”, J Chromatogr A, Vol. 1216, pp. 1008－1018, 2009
Bier M., Palusinski O. A., Mosher R. A. and Saville, D. A., Science., “ Electrophoresis–Mathematical–Modeling and Computer–Simulation”, Vol. 219, pp. 1281－1287, 1983
Cattaneo C., “A form of heat conduction equation which eliminates the paradox of instantaneous propagation”, Compute Rendus 247, pp. 431－433, 1958
Chang S. C., “The method of space-time conservation element and solution element – A new approach for solving the Navier-Stokes and Euler equations”, J. Comput. Phys, Vol. 119, pp. 295－324, 1995
Chang, S. C., “Courant number insensitive CE/SE schemes”, AIAA paper. 2002, 2002－3890
Chatterjee A., “Generalized numerical formulations for multi-physics microfluidics-type applications”, J. Micromech. Microeng., Vol. 13, pp. 758－767, 2003
Cui H., Dutta P. and Ivory C. F., “Isotachophoresis of proteins in a networked microfluidic chip: Experiment and 2-D simulation”, Electrophoresis, Vol. 28, pp. 1138－1145, 2007
Cui H., Horiuchi K., Dutta P. and Ivory C. F., “Isoelectric focusing in a poly(dimethylsiloxane) microfluidic chip”, Anal. Chem., Vol. 77, pp. 1303－1309, 2005
Cui H., Horiuchi K., Dutta P. and Ivory C. F., “Multistage isoelectric focusing in a polymeric microfluidic chip”, Anal. Chem., Vol. 77, pp. 7878－7886, 2005
Dose E. V. and Guiochon G. A., “High-resolution Modeling of Capillary Zone Electrohoresis and Isotachophoresis”, Anal. Chem., Vol. 63, pp. 1063－1072, 1991
Ermakov S. V., Mazhorova O. S. and Zhukov M. Y., “Computer simulation of transient states in capillary zone electrophoresis and isotachophoresis”, Electrophoresis, Vol. 13, pp. 838－848, 1992
Ermakov S. V., Bello M. S. and Righetti, P. G., “Numerical algorithms for capillary electrophoresis”, J. Chromatogr. A, Vol. 661, pp. 265－278, 1994
Gebauer P., Malá Z. and Boček P., “Recent progress in capillary ITP”, Electrophoresis, Vol. 28, pp. 26－32, 2007
Hrukša V., Jaroš M. and Gaš B., “Simul 5 - Free dynamic simulator of electrophoresis”, Electrophoresis, Vol. 27, pp. 984－991, 2006
Huang W., “Practical Aspects of Formulation and Solution of Moving Mesh Partial Differential Equations”, J. Comput. Phys., Vol. 171, pp. 753－775, 2001
Ikuta N. and Hirokawa T., “Numerical simulation for capillary electrophoresis I. Development of a simulation program with high numerical stability”, J. Chromatogr. A, Vol. 802, pp. 49－57, 1998
Jin C. and Xu K., “An adaptive grid method for two-dimensional viscous flows”, J. Comput. Phys., Vol. 218, pp. 68－81, 2006
Křivánková L., Pantůčková P. and Boček P., “Isotachophoresis in zone electrophoresis”, J. Chromatogr. A, Vol. 838, pp. 55－70, 1999
Loh C. Y., Hultgren L. S. and Chang S. C., “Wave computation in compressible flow using space-time conservation element and solution element method”, AIAA journal, Vol. 39, pp. 794－801, 2001
Martens J. H. P. A., Reijenga J. C., ten Thije Boonkkamp J. H. M., Mattheij R. M. M. and Everaerts F. M., “Transient modelling of capillary electrophoresis Isotachophoresis”, J. Chromatogr. A, Vol. 772, pp. 49－62, 1997
Mosher R. A., Thormann W. and Bier M., “An explanation for the Plateau phenomenaon in Isoelectric-Focusing”, J. Chromatogr., Vol. 351, pp. 31－38, 1986
Mosher R. A., Dewey D., Thormann W., Saville D. A. and Bier M., “Computer-Simulation and Experimental validation of the Electrophoretic Behavior of Proteins”, Anal. Chem., Vol. 61, pp. 362－366, 1989
Mosher R. A. and Thormann W., “Experimental and Theoretical Dynamics of Isoelectric-Focusing 4. Cathodic, Anodic and Symmetrical Drifts of the pH gradient”, Electrophoresis, Vol. 11, pp. 717－723, 1990
Mosher R. A., Saville D. A. and Thormann W., “The Dynamics of Electrophoresis”, VCH, Wenhem, 1992.
Palusinski O. A., Graham A., Mosher R. A., Bier M. and Saville D. A., “ Theory of Electrophoretic separations 2. Construction of a Numerical-Simulation scheme and its Applications”, AIChE Journal., Vol. 32, pp. 215－223, 1986
Petr J., Maier V., Horáková J., Ševcík J. and Stránský Z., “Determination of some heavy metal cations in molten snow by transient isotachophoresis/capillary zone electrophoresis”, J. Sep. Sci., Vol. 29, pp. 2705－2715, 2006
Righetti P. G., “Electrophoresis: The march of pennies, the march of dimes”, J. Chromatogr. A, Vol. 1079, pp. 24－40, 2005
Saville D. A. and Palusinski O. A., “Theory of Electrophoretic separations 1. Formulation of a Mathematical-Model”, AIChE J., Vol. 32, pp. 207－214, 1986
Schwer C., Gaš B., Lottspeich F. and Kenndler E., “Computer simulation and experimental evaluation of on-column sample preconcentration in capillary zone electrophoresis by discontinuous buffer systems”, Anal. Chem., Vol. 65, pp. 2108－2115, 1993
Schafer-Nielsen C., “A computer model for time-based simulation of electrophoresis systems with freely defined initial and boundary conditions”, Electrophoresis, Vol. 16, pp. 1369－1376, 1995
Shim J., Dutta P. and Ivory C. F., “Modeling and simulation of IEF in 2-D microgeometries”, Electrophoresis, Vol. 28, pp.572－586, 2007
Shim S. H., Riaz A., Choi K. W. and Chung D. S., “Dual stacking of unbuffered saline samples, transient isotachophoresis plus induced pH junction focusing”, Electrophoresis, Vol. 24, pp. 1603－1611, 2003
Silvertand L. H. H., Torano J. S., van Bennekom W. P. and de Jong G. J., “Recent developments in capillary isoelectric focusing”, J. Chromatogr. A, Vol. 1204, pp. 157－170, 2008
Sounart T. L. and Baygents J. C., “Simulation of electrophoretic separations by the flux-corrected transport method”, J Chromatogr A, Vol. 890, pp. 321－336, 2000
Tang H. Z. and Tang T., “Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws”, SIM J. Numer. Anal., Vol. 41, pp. 487－515, 2003
Tang H. Z., “Solution of the shallow-water equations using an adaptive moving mesh method”, Int. J. Numer. Meth. Fluids, Vol. 44, pp. 789－810, 2004
Tang H. Z. and Tang T., “Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws”, SIM J. Numer. Anal., Vol. 41, pp. 487－515, 2003
Thormann W. and Mosher R. A., “High-resolution computer simulation of the dynamics of isoelectric focusing using carrier ampholytes: Focusing with concurrent electrophoretic mobilization is an isotachophoretic process”, Electrophoresis, Vol. 27, pp. 968－983, 2006
van Dam A. and Zegeling P. A., “A robust moving mesh finite volume method applied to 1D hyperbolic conservation laws from magnetohydrodynamics”, J. Comput. Phys., Vol. 216, pp. 526－546, 2006
Yu S. T. and Chang S. C., “Treatments of stiff source terms in conservation laws by the method of space-time conservation element and solution element”, AIAA 97-0435, 1997
Yu J. W., Chou Y. and Yang R. J., “High-resolution modeling of isotachophoresis and zone electrophoresis”, Electrophoresis, Vol. 29, pp. 1048－1057, 2008
Zhang Z. C., Yu S. T. and Chang S. C., “A space-time conservation element and solution element method for solving the two- and three-dimension unsteady euler equations using quadrilateral and hexahedral meshes”, J Comput Phys, Vol. 175, pp. 168－199, 2002