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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} }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.