Study On Energy-preserving Extrapolation Algorithms For Hamiltonian Systems

Almost all real and dissipative negligible physical processes can be written as Hamiltonian systems,and therefor such systems widely exist in classical mechanics,astrophysics,mechanics of materials and so on.Energy conservation is an important feature of Hamiltonian systems.Whether it can be effectively preserved in numeri-cal simulation has become one of the criteria for evaluating the quality of numerical simulation.Therefore,this thesis studies energy-preserving algorithms for Hamiltoni-an systems,aiming at solving(or partially solving)the challenge of constructing and implementing high-order energy-preserving algorithms.Extrapolation is a post-processing technique for numerical simulation.By com-bining the results of low-order algorithm,high-order numerical approximation can be obtained.Its advantage is that it is simple in form and easy to implement.Given a low-order numerical algorithm,the idea of the classical extrapolation method is to combine the approximate solutions obtained by different step sizes linearly,and to obtain more accurate numerical solutions by reasonably choosing the parameters.However,even if this low-order algorithm is energy-preserving,the new numerical schemes con-structed by classical extrapolation method are generally no longer energy-preserving.Therefore,we consider the so-called active extrapolation method to construct energy-preserving extrapolation method.The basic idea is to keep the compatibility condition of parameters when constructing extrapolation algorithm,and to replace the equation for high precision requirement with the equation for energy conservation requirement.We do this to construct a new energy-preserving extrapolation scheme.For systems with two invariants,an extrapolation algorithm for preserving both invariants can be constructed similarly.The numerical accuracy of the energy-preserving extrapolation method is ana-lyzed for a finite dimensional linear Hamiltonian system.Here the explicit Runge-Kutta(RK)method is chosen as the low-order numerical algorithm.We find that for a linear Hamiltonian system,the numerical accuracy of the energy-preserving extrap-olation method is related to the parity of the order of the RK method.When the order of the RK method is odd,the numerical accuracy will be improved by one order.But if the order of the RK method is even,the numerical accuracy will remain unchanged.This conclusion is only valid for linear Hamiltonian systems.For nonlinear Hamilto?nian systems,we got some other interesting conclusions.We test the numerical simulation results of energy-preserving extrapolation method on the harmonic oscillator model and the Kepler problem which are examples of linear and nonlinear Hamiltonian system respectively.And we test the numerical simulation results of energy-preserving extrapolation method on the Schrodinger e-quation which is an example of partial differential equations.Here we still choose the explicit RK method as the low-order numerical algorithm.In terms of energy error,the energy-preserving extrapolation method has a good energy-preserving effect,and the energy error is smaller than that of AVF method.Besides,we test the conver-gence order.For the harmonic oscillator model,it is found that the energy-preserving extrapolation scheme can improve the convergence order for the odd-order explicit RK methods,but for the even-order explicit RK methods,the accuracy of the numer-ical schemes remains unchanged.Because the harmonic oscillator model is a linear Hamiltonian system,the experimental results are consistent with the theoretical anal-ysis.For the Kepler problem and the Schrodinger equation,the convergence order of energy-preserving extrapolation method is still related to the parity of the convergence order of RK method.In addition,for systems with two first integrals,such as the Ke-pler problem and the Schrodinger equation,an extrapolation scheme is constructed to simultaneously preserve both first integrals,and the numerical accuracy is also related to the parity of the convergence order of RK method.