Applications Of The Symplectic Methods To Hyperbolic Partial Differential Equations

Numerical calculations have been mainly used on the research of non-integrablesystems. However，the traditional numerical algorithms appear the deviation of theirtrajectry which are caused by their artificial dissipation. Therefore，symplecticalgorithms take more and more attraction because of the advantage on the stabilityand symplectic structure.The second-order symplectic method，as the simplest of the explicit symplecticintegrators，otfen be used to construct high order symplectic integrators. Yoshidapropose a method to construct the higher order integrators through by the combinationof lower order integrators. Li and Wu construct two new three stage fourth-ordersymplectic integrators which have the similar form with the second-order integratorsand high numerical accuracy.This paper mainly discusses the wave equation and Sine-Gordon equation asexamples to study the application of symplectic algorithms to hyperbolic partialdifferential equations. Firstly，we get finite-dimensional Hamilton systems through bythe numerically discrete on the spatial orientation of PDEs. Then we use symplecticmethods to solve the Hamiltonian system. So we construct the semi-discretesymplectic methods to the wave equation. Also we disscuss the stability of thesesemi-discrete methods which are constructed by different numerically discrete formsand symplectic methods to the wave equation. As a point，the Hamilton system hasa quadratic form of the variational position variables. Therefor，we use the new threestage fourth-order symplectic integrators to solve the equation through this method，that we get high order semi-discrete methods which can be widely applied tohyperbolic wave equation. In addition，this article discusses the multi-symplecticmethods by composite construction. We obtain individually superiority aboutaccuracy and conputational eiffency through the numerical comparison between thesetwo methods to nonlinear systems.