Abstract
In this paper, we develop and analyze the energy conservative time high-order AVF compact finite difference methods for variable coefficient acoustic wave equations in two dimensions. We first derive out an infinite-dimensional Hamiltonian system for the variable coefficient wave equations and apply the spatial fourth-order compact finite difference operator to the equations of the system to obtain a semi-discrete approximation system, which can be cast into a canonical finite-dimensional Hamiltonian form. We then apply the second-order and fourth-order AVF techniques to propose the fully discrete energy conservative time high-order AVF compact finite difference methods for wave equations in two dimensions. We prove that the proposed semi-discrete and fully-discrete schemes satisfy energy conservations in the discrete forms. We further prove that the semi-discrete scheme has the fourth-order convergence order in space and the fully-discrete AVF compact finite difference method has the fourth-order convergence order in both time and space. Numerical tests confirm the theoretical results.
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Acknowledgements
This work was supported partially by Natural Sciences and Engineering Research Council of Canada and by Natural Science Foundation of China under Grant 11271232. B. Hou would also thank the Department of Mathematics and Statistics at York University for her visit.
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Hou, B., Liang, D. & Zhu, H. The Conservative Time High-Order AVF Compact Finite Difference Schemes for Two-Dimensional Variable Coefficient Acoustic Wave Equations. J Sci Comput 80, 1279–1309 (2019). https://doi.org/10.1007/s10915-019-00983-6
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DOI: https://doi.org/10.1007/s10915-019-00983-6
Keywords
- AVF method
- Compact finite difference
- Energy conservation
- Convergence analysis
- Variable coefficient wave equation
- Hamiltonian