A Meshless Symplectic Algorithm Based On Radial Basis Functions

Recently, as both the radial basis function method and symplectic scheme are pow-erful tools to solve differential equations numericaly. In this Ph.D thesis, we will proposed a meshless symplectic scheme based on radial basis function theory.Radial basis function plays an important role in fields of soving partial differential equations(PDEs) numericaly, fitting of the scattered data, and signal processing. For years, The theory of solving PDEs by radial basis function method has been developed and improved. Since the method by using radial basis function do not require to have any pre-orocess of meshing, it is easier to implement.The symplectic schemes are the algotithm which can preserve the intrinsical character (symplectic structure) of the Hamiltonian PDEs, they possessed a good property in long-term tracking capability.To develop a meshless symplectic algorithm or meshless energy-conserving numerical scheme on scattered nodes based on radial basis theory motivates the current work.All the physical courses whose dissipative effects are negligible can be expressed as Hamiltonian systems. The Hamiltonian system is universal in the nature, most of soliton equations can be treated as Hamiltonian system(in finite dimensions). Hamiltonain system has the important property of being area-preserving(symplectic). The basic principle of modern numerical computation is to preserve the intrisical character of the original problems at every time step. Therefor, it is necessary to study numerical methods which preserve the symplectic structure of the Hamiltonian system. Academician Kang Feng, the founder of Chiness computational mathematics, put forward the symplectic geometric algorithm systemically which preserve symplectic structure of the Hamiltonian system in1984. Symplectic integrator became a hot quetion for discussion in computational schience consequently in domestic and foreign, and numbers of related achievements welled up.The structure of this thesis is as follows:In Chaper1, we introduce the background of symplectic algorithms, and review the preliminary about the Hamiltonian and symplectic geometry.In Chapter2, the commonly used methods to construct symplectic algorithms are recalled, including the generating function method, the implicit Runge-Kutta methods, the explicit method for the-geparable Hamiltonian system.In Chapter3, we introduce the standard method to demote the Hamiltonian PDEs into a finite-dimensional Hamiltonian system. We also introduce how a space discretization by using finite difference and spline function method. It’s worth noting that there is few literatures about constucting symplectic integrators by using spline function method, it is also the starting point of innovation which set a ateppingstone to constructing symplectic integrators by using radial basis function method. Finally, we introduce a common method to constructing energy-conerving scheme.In Chapter4, We introduce the theory about radial basis function quasi-interpolation and interpolation. A novel discrete formulas will be introduced for the integration of a function, with a radial basis kernel method. And its error is also analysed, which is prepared for error anlysis of symplectic radial basis functions algorithm.In Chapter5, We study symplectic algorithm of one-dimensinal Hamiltonian wave equation by using radial basis function method. We will introduce the method to consctuct symplectic scheme with radial basis quasi-interpolation and interpolation. Theoretical analysis including error estimates of convergence and properties of long-time tracking capability are shown in this paper. Numerical examples verified the effectiveness of the method.in Chapter6, We study symplectic algorithm for Hamiltinian system with multi-variate by using radial basis function interpolation method, we will discuss how a space discretization by using RBFs method, makes that the space discretized system also has some invariants which well approximate the energy of the system. There are two method to discrete the problem at first in space dimension. One is discrete the equation by radial basis approximation and find the coincide discrete energy. The other one is discrete the energy by radial basis approximation and find the coincide equations. We analysis the two methods in theory and campare them in numerical test.In the last Chapter, we summarize the main conclusions of the dissertation. The future propects and some challenge as well as some open problem are list in the end.