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 - <p style="text-align: justify;">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.</p>