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Volume 40, Issue 4
A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation

Jialing Wang & Yali He

DOI: 10.4208/cmr.2024-0018

Commun. Math. Res., 40 (2024), pp. 413-436.

Published online: 2024-12

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  • Abstract

In this paper, a meshless energy-preserving algorithm which can be arbitrarily high-order in temporal direction for the beam equation has been proposed. Based on the method of lines, we first use the radial basis function quasi-interpolation method to discretize spatial variable and obtain a semi-discrete Hamiltonian system by using the premultiplication of a diagonal matrix. Then, symplectic Runge-Kutta method that can conserve quadratic invariants exactly has been used to discretize the temporal variable, which yields a fully discrete meshless scheme. Due to the specific quadratic energy expression of the beam equation, the proposed meshless scheme here is not only energy-preserving but also arbitrarily high-order in temporal direction. Besides uniform and nonuniform grids, numerical experiments on random grids are also conducted, which demonstrate the properties of the proposed scheme very well.

  • Keywords

Beam equation, meshless scheme, energy-preserving method, radial basis function, symplectic Runge-Kutta method.

  • AMS Subject Headings

37K05, 65L06, 65L12, 65N40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-40-413, author = {Wang , Jialing and He , Yali}, title = {A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation}, journal = {Communications in Mathematical Research }, year = {2024}, volume = {40}, number = {4}, pages = {413--436}, abstract = {

In this paper, a meshless energy-preserving algorithm which can be arbitrarily high-order in temporal direction for the beam equation has been proposed. Based on the method of lines, we first use the radial basis function quasi-interpolation method to discretize spatial variable and obtain a semi-discrete Hamiltonian system by using the premultiplication of a diagonal matrix. Then, symplectic Runge-Kutta method that can conserve quadratic invariants exactly has been used to discretize the temporal variable, which yields a fully discrete meshless scheme. Due to the specific quadratic energy expression of the beam equation, the proposed meshless scheme here is not only energy-preserving but also arbitrarily high-order in temporal direction. Besides uniform and nonuniform grids, numerical experiments on random grids are also conducted, which demonstrate the properties of the proposed scheme very well.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2024-0018}, url = {http://global-sci.org/intro/article_detail/cmr/23700.html} }
TY - JOUR T1 - A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation AU - Wang , Jialing AU - He , Yali JO - Communications in Mathematical Research VL - 4 SP - 413 EP - 436 PY - 2024 DA - 2024/12 SN - 40 DO - http://doi.org/10.4208/cmr.2024-0018 UR - https://global-sci.org/intro/article_detail/cmr/23700.html KW - Beam equation, meshless scheme, energy-preserving method, radial basis function, symplectic Runge-Kutta method. AB -

In this paper, a meshless energy-preserving algorithm which can be arbitrarily high-order in temporal direction for the beam equation has been proposed. Based on the method of lines, we first use the radial basis function quasi-interpolation method to discretize spatial variable and obtain a semi-discrete Hamiltonian system by using the premultiplication of a diagonal matrix. Then, symplectic Runge-Kutta method that can conserve quadratic invariants exactly has been used to discretize the temporal variable, which yields a fully discrete meshless scheme. Due to the specific quadratic energy expression of the beam equation, the proposed meshless scheme here is not only energy-preserving but also arbitrarily high-order in temporal direction. Besides uniform and nonuniform grids, numerical experiments on random grids are also conducted, which demonstrate the properties of the proposed scheme very well.

Wang , Jialing and He , Yali. (2024). A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation. Communications in Mathematical Research . 40 (4). 413-436. doi:10.4208/cmr.2024-0018
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