Abstract
It has been demonstrated that spectral deferred correction (SDC) methods can achieve arbitrary high order accuracy and possess good stability properties. There have been some recent interests in using high-order Runge-Kutta methods in the prediction and correction steps in the SDC methods, and higher order rate of convergence is obtained provided that the quadrature nodes are uniform. The assumption of the use of uniform mesh has a serious practical drawback as the well-known Runge phenomenon may prevent the use of reasonably large number of quadrature nodes. In this work, we propose a modified SDC methods with high-order integrators which can yield higher convergence rates on both uniform and non-uniform quadrature nodes. The expected high-order of accuracy is theoretically verified and numerically demonstrated.
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Acknowledgements
The first author thanks the useful discussions with Professors Chi-Wang Shu, John Strain and Jing-Mei Qiu. We are also grateful for Prof. Qiu for kindly providing us some relevant codes. This research was supported by Hong Kong Research Grants Council and Hong Kong Baptist University.
The second author is supported by the Croucher Foundation of Hong Kong, the National Nature Science Foundation of China (11001259), the National Center for Mathematics and Interdisciplinary Science, CAS and the President Foundation of AMSS-CAS.
The third author is supported by the National Science Foundation of China (11126215, 11201167).
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Tang, T., Xie, H. & Yin, X. High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes. J Sci Comput 56, 1–13 (2013). https://doi.org/10.1007/s10915-012-9657-9
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DOI: https://doi.org/10.1007/s10915-012-9657-9