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Fourier Spectral Methods for Degasperis–Procesi Equation with Discontinuous Solutions

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Abstract

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 DP equation are provided to illustrate the accuracy and capability of the methods.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei , 230026, Anhui, P.R. China

    Yinhua Xia

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  1. Yinhua Xia
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Correspondence to Yinhua Xia.

Additional information

Research supported by NSFC Grant No.11101400, No.11371342, Doctoral Fund of Ministry of Education of China No. 20113402120015 and SRF for ROCS SEM.

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Xia, Y. Fourier Spectral Methods for Degasperis–Procesi Equation with Discontinuous Solutions. J Sci Comput 61, 584–603 (2014). https://doi.org/10.1007/s10915-014-9839-8

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  • DOI: https://doi.org/10.1007/s10915-014-9839-8

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