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Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations |
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| Authors: | Cao Waixiang Li Dongfang Zhang Zhimin |
| Institution: | 1.School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China ;2.School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China ;3.Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China ;4.Beijing Computational Science Research Center, Beijing, 100193, China ;5.Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA ; |
| Abstract: | In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without certain time-step restrictions. The key is to estimate directly the solution bounds in the H2-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence and optimal error estimates, respectively, of the proposed fully discrete schemes. |
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