Fourier spectral methods for Degasperis-Procesi equation with discontinuous solutions

Yinhua Xia University of Science and Technology of China

Numerical Analysis and Scientific Computing mathscidoc:1612.25008

Journal of Scientific Computing, 61, 584-603, 2014.6
In this paper, we develop, analyze and test the Fourier spectral methods for solving the Degasperis-Procesi(DP) equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The $L^2$ stability is obtained for general numerical solutions of the Fourier Galerkin method and Fourier collocation (pseudospectral) method.By applying the Gegenbauer reconstruction technique as a post-processing method to the Fourier spectral solution, we reduce the oscillations arising from the discontinuity successfully. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the methods.
Degasperis-Procesi equation, discontinuous solution, Fourier Galerkin method, Fourier collocation method, $L^2$ stability, Gegenbauer reconstruction.
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@inproceedings{yinhua2014fourier,
  title={Fourier spectral methods for Degasperis-Procesi equation with discontinuous solutions},
  author={Yinhua Xia},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161231220107748964704},
  booktitle={Journal of Scientific Computing},
  volume={61},
  pages={584-603},
  year={2014},
}
Yinhua Xia. Fourier spectral methods for Degasperis-Procesi equation with discontinuous solutions. 2014. Vol. 61. In Journal of Scientific Computing. pp.584-603. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161231220107748964704.
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