Convergence analysis of a linearized Crank–Nicolson scheme for the two‐dimensional complex Ginzburg–Landau equation |
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| Authors: | Ya‐nan Zhang Zhi‐zhong Sun Ting‐chun Wang |
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| Institution: | 1. School of Mathematics Sciences, Soochow University, Suzhou 215006, Peoples Republic of China;2. Department of Mathematics, Southeast University, Nanjing 210096, Peoples Republic of China;3. College of Math & Physics, Nanjing University of Information Science & Technology, Nanjing 210044, Peoples Republic of China |
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| Abstract: | A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L2 ‐norm by the discrete energy method. The convergence order is \begin{align*}\mathcal{O}(\tau^2+h_1^2+h^2_2)\end{align*}, where τ is the temporal grid size and h1,h2 are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 |
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| Keywords: | complex Ginzburg– Landau equation convergence discrete Sobolev's inequality energy method finite difference |
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