Energy-preserving Numerical Methods

Nonlinearity generally exists in mathematical models of physic and engineering problems, and it is more difficult to find the exact solutions of nonlinear problems. Therefore, it is of great significance to study the numerical solutions for the models.A basic idea behind the design of the numerical scheme is that they can preserve the properties of the original problems as much as possible. A variety of structure-preserving algorithms are produced based on this thought. The mathematical framework of the Hamiltonian system is symplectic geometry, and Feng Kang systematically studied the symplectic algorithm based on symplectic geometry systems. The algorithm which can preserve the symplectic structure is called the symplectic algorithm. The Hamilton partial differential equations have multi-symplectic conservation laws, the methods which preserve the multi-symplectic conservation laws are known as the multi-symplectic methods. The methods for the lie-group differential equations which preserve the lie-group structure are called the lie-group methods. The structure-preserving thought is also applied to highly-oscillatory differential equations.We hope that the numerical methods can preserve the energy functions as much as possible for energy-preserving systems. The numerical methods which preserve the energy functions are called energy-preserving methods. Quispel, Mclachlan, Robidou and Gonzalez have carried out a detailed study of the discrete gradient methods.In this report, we introduced three representation of classical mechanics from movement of pendulum to particularly recount and prove the hamilton systems and its property, symplectic geometric algorithms, symmetric numerical methods and energy-momentum methods. Then we introduce the energy-momentum methods, we first prove that the implicit midpoint method preserve quadratic invariant, then we show the modified implicit midpoint method can preserve both the momentum and energy. At the same time, we give the definition of discrete gradient and introduce some discrete gradient methods.We mainly discuss one kind of Hamilton function which take the form H=fT Bf, B=BT. We give a kind of discrete gradient, and study the symmetric properties. A concrete example shows that numerical method given by our discrete gradient has a simpler form.