Volume 11, Issue 1
A Splitting Method for the Degasperis--Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method

Yunrui Guo, Wenjing Yang, Hong Zhang, Ji Wang & Songhe Song

Adv. Appl. Math. Mech., 11 (2019), pp. 53-71.

Published online: 2019-01

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  • Abstract

The Degasperis--Procesi (DP) equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use an optimized finite difference weighted essentially non-oscillatory (OWENO) scheme. New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution to the DP equation, which evidently reduces the dissipation arising from discontinuities simultaneously removing nonphysical oscillations. For the elliptic equation, the Fourier pseudospectral method (FPM) is employed to discretize the high order derivative. Due to the combination of the WENO reconstruction and FPM, the splitting method shows an excellent performance in capturing the formation and propagation of shockpeakon solutions. The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.

  • Keywords

Degasperis--Procesi equation discontinuous solution weighted essentially non-oscillatory method pseudospectral method.

  • AMS Subject Headings

65M10 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-11-53, author = {Yunrui Guo, Wenjing Yang, Hong Zhang, Ji Wang and Songhe Song}, title = {A Splitting Method for the Degasperis--Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {1}, pages = {53--71}, abstract = {

The Degasperis--Procesi (DP) equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use an optimized finite difference weighted essentially non-oscillatory (OWENO) scheme. New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution to the DP equation, which evidently reduces the dissipation arising from discontinuities simultaneously removing nonphysical oscillations. For the elliptic equation, the Fourier pseudospectral method (FPM) is employed to discretize the high order derivative. Due to the combination of the WENO reconstruction and FPM, the splitting method shows an excellent performance in capturing the formation and propagation of shockpeakon solutions. The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0054}, url = {http://global-sci.org/intro/article_detail/aamm/12921.html} }
TY - JOUR T1 - A Splitting Method for the Degasperis--Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method AU - Yunrui Guo, Wenjing Yang, Hong Zhang, Ji Wang & Songhe Song JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 53 EP - 71 PY - 2019 DA - 2019/01 SN - 11 DO - http://dor.org/10.4208/aamm.OA-2018-0054 UR - https://global-sci.org/intro/aamm/12921.html KW - Degasperis--Procesi equation KW - discontinuous solution KW - weighted essentially non-oscillatory method KW - pseudospectral method. AB -

The Degasperis--Procesi (DP) equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use an optimized finite difference weighted essentially non-oscillatory (OWENO) scheme. New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution to the DP equation, which evidently reduces the dissipation arising from discontinuities simultaneously removing nonphysical oscillations. For the elliptic equation, the Fourier pseudospectral method (FPM) is employed to discretize the high order derivative. Due to the combination of the WENO reconstruction and FPM, the splitting method shows an excellent performance in capturing the formation and propagation of shockpeakon solutions. The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.

Yunrui Guo, Wenjing Yang, Hong Zhang, Ji Wang & Songhe Song. (1970). A Splitting Method for the Degasperis--Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method. Advances in Applied Mathematics and Mechanics. 11 (1). 53-71. doi:10.4208/aamm.OA-2018-0054
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