Study On Two Kinds Of Conformal Symplectic Numerical Methods

With the development of technology and society,it is found that differential equations become more and more important.But it is very difficult to find the exact solution of the corresponding equation directly for the deterministic differential equation and stochastic differential equation with strong nonlinearity and coupling.Therefore,the construction of appropriate and useful numerical methods has become an important issue for scholars.As an important part of the numerical method of differential equation,the essence of structure preserving algorithm is that it can keep some structures of the original equation as much as possible,such as symplectic structure,energy conservation,etc.Since most systems are not conservative in real life,this paper studies the preservation of conformal symplectic structure and conformal quadratic invariants of exponential Runge-Kutta-Nystr(?)m method and stochastic partitioned exponential Runge-Kutta method for differential equations with dissipation term,namely damped differential equations.In the first and second part,it mainly introduces the background,purpose,significance and research status of numerical methods for differential equations at home and abroad,and briefly introduces the related basic knowledge and definitions.In the third part,for the second order damped ordinary differential equation,we mainly study a kind of the exponential Runge-Kutta-Nystr(?)m method which can keep the conformal symplectic structure,and give the sufficient conditions which can keep the conformal symplectic structure and the conformal quadratic invariant.Then,the theory is verified by numerical experiments.In the fourth part,for damped stochastic differential equations,the numerical scheme of stochastic partitioned exponential Runge-Kutta method is constructed,the method is studied to keep the conformal symplectic structure and the conformal quadratic invariant,and the convergence of a special kind of stochastic partitioned exponential Runge-Kutta method is analyzed.According to the symplectic sufficient condition and convergence order condition of numerical method,several kinds of stochastic partitioned exponential Runge-Kutta methods with mean square one order convergence are constructed.Finally,numerical examples are used to verify the effectiveness of the theoretical results.