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標題: 以龍格-庫塔耐斯壯和分部求和差分法求解波方程
Runge-Kutta-Nyström and Summation by Parts Difference Method for Wave Equation
作者: 周柔廷
Jou-Ting Chou
關鍵字: 微分矩陣;Runge-Kutta-Nyström 法;補償法;SBP Operator;Runge-Kutta-Nyström method;penalty method
引用: [1] Bengt Fornberg, The pseudospectral method:Comparisons with finite differences for the elastic wave equation, Geophysics, Vol. 52, No. 4, April 1987, 483-501. [2] E. Doedel, Finite difference methods for nonlinear two-point boundary-value problem, SIAM Journal of Numerical Analysis 16, 1979, 173–185. [3] Kurt J. Marfurt, Accuracy of finite‐difference and finite‐element modeling of the scalar and elastic wave equations, Geophysics, Vol. 49, No. 5, May 1984, 533-549. [4] E. Mossberg, Higher order finite difference methods for wave propagation problems, IT Licentiate thesis, Department of Information Technology, Uppsala University, Oppsala, Sweden, 2002. [5] GD Smith, Numerical solution of partial differential equations: finite difference methods, 1985. [6] Gregory Fredricks, Roger B. Nelson, Summation by Parts, The College Mathematics Journal, Vol. 23, No. 1, Jan 1992, 39-42. [7] Ken Mattsson, Jan Nordström, Summation by parts operators for finite difference approximations of second derivatives, Journal of Computational Physics 199, 2004, 503-540. [8] Ken Mattsson, Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients, Journal of Scientific Computing, Vol. 51, June 2012, 650-682. [9] D. Funaro, D. Gottlieb, A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations, Mathematics of Computation, Vol. 51, No. 184 , Oct 1988, 599–613. [10] Özgür Yeniay, Penalty function methods for constrained optimization with genetic algorithms, Mathematical and Computational Applications, Vol. 10, No. 1, 2005, 45-56. [11] Pelle Olsson, Summation by Parts, Projections, and Stability, I. Mathematics of Computation, Vol. 64, No. 211, July 1995, 1035-1065. [12] Ken Mattsson, Frank E Ham, Gianluca Iaccarino, Stable and accurate wavepropagation in discontinuous media, Journal of Computation Physics 227, 2008, 8753-8767. [13] Heinz-Otto Kreiss, N. Anders Petersson, Jacob Ystrom, Difference approximations for the second order wave equation, SIAM J. Numer. Anal. Vol. 40, No. 5, 2002, 1940–1967. [14] Bertil Gustafsson, Per Wahlund, Time compact difference methods for wave propagation in discontinuous media, SIAM J. Sci. Comput. Vol. 26, No. 1, 2004, 272-293. [15] Bertil Gustafsson, Eva Mossberg, Time compact high order difference methods for wave propagation, SIAM J. Sci. Comput. Vol. 26, No. 1, 2004, 259-271. [16] Chun-Hao Teng, Misun Min, Juen-Kai Wang, Pseudospectral and Runge-Kutta-Nyström methods for second-order wave equations: Stable and accurate boundary treatments. [17] Mark H. Carpenter, David Gottlieb, Saul Abarbanel, Wai-Sun Don, The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error, SIAM J. Sci. Comput. Vol. 26, No. 6, 1995, 1241-1252.
本論文討論在邊界條件為零和邊界條件不為零的情況下求解波動方程式,我們使用滿足有限差分法的微分運算子和Runge-Kutta-Nyström 法加上補償法建置計算格式,接著透過撰寫程式進行測試,其所得數值解結果精確且隨

In this thesis, we compute the approximate solution of the wave equation in two different boundary conditions. One is zero and the other is not. We use SBP operators and construct the computation scheme by Runge-Kutta-Nyström method and penalty method. Then we test the cases by the program and the result of the approximate solution is accurate and converge by refining grid.
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