A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation (Communications in Mathematical Research, 2024)
In this paper, a meshless energy-preserving algorithm which can bearbitrarily high-order in temporal direction for the beam equation has beenproposed. 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 yieldsa fully discrete meshless scheme. Due to the specific quadratic energy expression of the beam equation, the proposed meshless scheme here is not onlyenergy-preserving but also arbitrarily high-order in temporal direction. Besides uniform and nonuniform grids, numerical experiments on random gridsare also conducted, which demonstrate the properties of the proposed schemevery well.