The Research Of Energy-Preserving Numerical Methods For Hamilton Equations

Hamilton system has a wide range of applications in astrophysics,quantum mechanics and other fields.All real physical processes with negligible dissipation can always be cast in suitable Hamiltonian form.However,for many Hamilton equations,it is often very difficult to find the exact solutions.Therefore,it is of theoretical and practical significance to systematically study the numerical solutions of Hamilton equations.We hope that numerical methods are able to preserve the important properties of Hamilton equations as much as possible.Discrete gradients play a very important role in constructing energy-preserving numerical methods for Hamilton equations.We can use discrete gradients to construct energy-preserving numerical methods,discrete gradients include coordinate increment discrete gradient,mean value discrete gradient,etc.In this paper,we introduce some properties of Hamiltonian equations and knowledge of symplectic geometry algorithms,and mainly study the energy-preserving numerical methods for Hamilton equations.We study the relations between some discrete gradients of Hamiltonian functions fg,fgh,and give their relational expressions.On the basis of that,a method is given to construct the second order energy-preserving numerical methods by the first order energy-preserving numerical methods.Finally,the numerical simulation is carried out with a concrete example,the numerical results show that the new methods are the energy-preserving numerical methods.