Volume 19, Issue 3
Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation

Ching-Hao Yu & Tony Wen-Hann Sheu

Commun. Comput. Phys., 19 (2016), pp. 603-631.

Published online: 2018-04

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  • Abstract

In this paper a three-step scheme is applied to solve the Camassa-Holm (CH) shallow water equation. The differential order of the CH equation has been reduced in order to facilitate development of numerical scheme in a comparatively smaller grid stencil. Here a three-point seventh-order spatially accurate upwinding combined compact difference (CCD) scheme is proposed to approximate the first-order derivative term. We conduct modified equation analysis on the CCD scheme and eliminate the leading discretization error terms for accurately predicting unidirectional wave propagation. The Fourier analysis is carried out as well on the proposed numerical scheme to minimize the dispersive error. For preserving Hamiltonians in Camassa-Holm equation, a symplecticity conserving time integrator has been employed. The other main emphasis of the present study is the use of u−P−α formulation to get nondissipative CH solution for peakon-antipeakon and soliton-anticuspon head-on wave collision problems.

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@Article{CiCP-19-603, author = {}, title = {Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {19}, number = {3}, pages = {603--631}, abstract = {

In this paper a three-step scheme is applied to solve the Camassa-Holm (CH) shallow water equation. The differential order of the CH equation has been reduced in order to facilitate development of numerical scheme in a comparatively smaller grid stencil. Here a three-point seventh-order spatially accurate upwinding combined compact difference (CCD) scheme is proposed to approximate the first-order derivative term. We conduct modified equation analysis on the CCD scheme and eliminate the leading discretization error terms for accurately predicting unidirectional wave propagation. The Fourier analysis is carried out as well on the proposed numerical scheme to minimize the dispersive error. For preserving Hamiltonians in Camassa-Holm equation, a symplecticity conserving time integrator has been employed. The other main emphasis of the present study is the use of u−P−α formulation to get nondissipative CH solution for peakon-antipeakon and soliton-anticuspon head-on wave collision problems.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.290914.030615a}, url = {http://global-sci.org/intro/article_detail/cicp/11102.html} }
TY - JOUR T1 - Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation JO - Communications in Computational Physics VL - 3 SP - 603 EP - 631 PY - 2018 DA - 2018/04 SN - 19 DO - http://dor.org/10.4208/cicp.290914.030615a UR - https://global-sci.org/intro/article_detail/cicp/11102.html KW - AB -

In this paper a three-step scheme is applied to solve the Camassa-Holm (CH) shallow water equation. The differential order of the CH equation has been reduced in order to facilitate development of numerical scheme in a comparatively smaller grid stencil. Here a three-point seventh-order spatially accurate upwinding combined compact difference (CCD) scheme is proposed to approximate the first-order derivative term. We conduct modified equation analysis on the CCD scheme and eliminate the leading discretization error terms for accurately predicting unidirectional wave propagation. The Fourier analysis is carried out as well on the proposed numerical scheme to minimize the dispersive error. For preserving Hamiltonians in Camassa-Holm equation, a symplecticity conserving time integrator has been employed. The other main emphasis of the present study is the use of u−P−α formulation to get nondissipative CH solution for peakon-antipeakon and soliton-anticuspon head-on wave collision problems.

Ching-Hao Yu & Tony Wen-Hann Sheu. (2020). Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation. Communications in Computational Physics. 19 (3). 603-631. doi:10.4208/cicp.290914.030615a
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