Study Of Energy-preserving Algorithms For The Hénon-Heiles System And A Boussinesq-type System

It is known that two vital conservative properties of a Hamiltonian system are the symplectic structure and the energy conservation.Massive experiments indicate that a conservative scheme,capable of preserving the discrete symplectic structure or en-ergy,plays a better role in numerical stability and long-term computational accuracy than those non-conservative.However,in the sense of B-series method,there exists no numerical algorithm capable of preserving both the symplectic structure and the energy for an arbitrary Hamiltonian system.This thesis only take the energy conser-vation into account to propose numerical schemes for the Henon-Heiles system and a Boussinesq-type system.The classic Henon-Heiles system,whose chaos phenomenon relates closely to the system energy,can be rewritten as a finite-dimensional Hamiltonian system.Whether the algorithm could preserve the energy dose make differences when ob-serving motional orbits.In the context,the averaged vector field method is applied and we obtain an energy-preserving scheme for the Henon-Heiles system.We take advantage of Poincare cuts to investigate the chaos and order.Numerical results re-veal that the chaos concerns with the choices of initial status apart from the system energy.In comparison with the energy-preserving algorithm,experiments based up-on a symplectic algorithm,derived from the implicit midpoint method,are carried out too.Numerical results demonstrate that both algorithms have good performances in simulations of the chaos and order.In addition,the energy-preserving algorithm conserves the energy of the Hdnon-Heiles system superiorly and larger time steps are workable.The generic Boussinesq system including 4 parameters is also called abcd-Boussinesq system.We merely study the b=b d type,which contains many water wave models such as the Bona-Smith system and the coupled BBM system.This Boussinesq-type system is energy-conservative because it can be rewritten as an infinite-dimensional Hamiltonian system.Whether the algorithm could preserve the energy dose make differences when observing the numerical wave propagation.We propose an energy-preserving algorithm for this Boussinesq-type system by applying the Fourier pseudo-spectral method to spatial discretization and the averaged vector field method to temporal discretization.In comparison with the energy-preserving al-gorithm,experiments based upon a symplectic algorithm,derived from the midpoint method,are carried out too.Numerical results indicate that both algorithms behave well in simulations of solitary wave propagations.In addition,the energy-preserving algorithm proves to be superior at conserving the energy of that Boussinesq-type sys-tem.