在本篇論文中，我們提出兩種解決一維度上非靜態反應-對流-擴散方程的高階緊緻有限差分法。對於空間的離散，第一種方法使用四階Spotz緊緻差分格式，而第二種方法則使用四階的指數型緊緻差分格式，至於時間變數之離散，兩者都使用Pade近似法。首先，我們先推導具來源項靜態方程的兩種四階緊緻差分法，接著將推導出的緊緻差分法應用到來源項為零的非靜態方程上，經此程序可以取得一個半離散形式，此半離散形式為一個大型的常微分方程式之初始值問題。最後，我們利用Pade近似法求取該初始值問題的數值近似解。在某些條件的假設之下，我們證明了這兩種方法都是無條件穩定。論文最後所提供的數值例子說明了新提出之緊緻差分方法是有效的，由這些數值結果，我們發現當網格佩克萊常數比較小時，這兩種差分方法對於空間以及時間變數都可達四階的精確度。然而，當網格佩克萊常數漸漸增大時，會使這兩種方法的數值解精確度惡化，而且在此情況下，第二個方法明顯比第一個方法來得精確。

In this thesis, we propose two high-order compact finite difference schemes for solving 1-D unsteady reaction-convection-diffusion problems. For the spatial discretization, the first scheme employs the fourth-order Spotz compact difference formula while the second scheme uses the fourth-order exponential compact difference formula. For discretizing the temporal variable, both schemes utilize the Pade approximation. First, we derive the spatially high-order compact difference schemes for the corresponding steady-state equation with a source term. We then apply the resulting compact difference schemes to the unsteady equation without source terms to obtain the semi-discrete formulation, which is an initial-value problem of a large system of ordinary differential equations. Finally, we apply the Pade approximation to compute the numerical solution of the initial-value problem. Under some assumptions, we prove that both schemes are unconditionally stable. Numerical examples are given to illustrate the effectiveness of the newly proposed compact difference schemes. From the numerical results, we find that for small mesh-Peclet numbers, both schemes achieve fourth-order accuracy in temporal and spatial variables. However, the accuracy of both schemes is deteriorated when the mesh-Peclet number is getting large, and in this case, the second scheme is apparently more accurate than the first scheme.

[1] H. Ding and Y. Zhang, A new difference scheme with high accuracy and absolute stability for
solving convection-diffusion equations, Journal of Computational and Applied Mathematics,
230 (2009), pp. 600-606.
[2] P.-W. Hsieh, S.-Y. Yang, and C.-S. You, A new high-accuracy compact finite difference
scheme for reaction-convection-diffusion problems with a small diffusivity, preprint, 2011.
[3] S. Karaa and J. Zhang, High order ADI method for solving unsteady convection-diffusion
problems, Journal of Computational Physics, 198 (2004), pp. 1-9.
[4] S. Karaa, A hybrid Pade ADI scheme of higher-order for convection-diffusion problems,
International Journal for Numerical Methods in Fluids, 64 (2010), pp. 532-548.
[5] K. W. Morton, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall,
London, UK, 1996.
[6] H.-G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differ-
ential Equations, Springer, New York, 1996.
[7] Z. F. Tian and S. Q. Dai, High-order compact exponential finite difference methods for
convection-diffusion type problems, Journal of Computational Physics, 220 (2007), pp. 952-
974.
[8] Z. F. Tian and Y. B. Ge, A fourth-order compact ADI method for solving two-dimensional
unsteady convection-diffusion problems, Journal of Computational and Applied Mathemat-
ics, 198 (2007), pp. 268-286.
[9] Z. F. Tian and P. X. Yu, A high-order exponential scheme for solving 1D unsteady convection-
diffusion equations, Journal of Computational and Applied Mathematics, 235 (2011), pp.
2477-2491.
[10] A. A. Salama and H. Z. Zidan, Fourth-order schemes of exponential type for singularly
perturbed parabolic partial differential equations, Rocky Mountain Journal of Mathematics,
36 (2006), pp. 1049-1068.
[11] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods,
Thrid Edition, Oxford University Press, Oxford, 1985.
[12] W. F. Spotz, High-Order Compact Finite Difference Schemes for Computational Mechanics,
Ph.D. Dissertation, the University of Texas at Austin, December 1995.