Abstract
By introducing appropriate intermediate variables and simplifying inhomogeneous terms of the traditional ADI schemes, an improved ADI scheme is proposed for linear hyperbolic equations and this ADI scheme is extended to nonlinear hyperbolic equations and compact ADI schemes in the present work. Meanwhile, the boundary and initial conditions are carefully analyzed to match the accuracy of these improved ADI schemes. Although the common (without compactification) and compact improved ADI schemes have the same accuracy with the corresponding traditional ADI schemes, respectively, it is not just a simple variant of ADI schemes for hyperbolic equations. Both theoretical analysis and numerical experiments show that compared with the traditional ADI scheme, the improved ADI scheme is more efficient for linear hyperbolic equations and more stable for nonlinear hyperbolic equations without loss of accuracy.
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Acknowledgements
The financial support from the National Natural Science Foundation of China (Grant Nos. 51531009 and 51474239), and Funds for International Cooperation and Exchange of the National Natural Science Foundation of China (Grant No. 51611130058) is greatly acknowledged. Authors would like to thank reviewers of this paper for their positive suggestions.
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Zhang, G., Cai, D. & Du, Y. Improved ADI Scheme for Linear Hyperbolic Equations: Extension to Nonlinear Cases and Compact ADI Schemes. J Sci Comput 72, 500–521 (2017). https://doi.org/10.1007/s10915-017-0366-2
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DOI: https://doi.org/10.1007/s10915-017-0366-2