Continuous Stage Stochastic Runge-Kutta Methods And Their Applications

Because of the universally existence of randomness in real life,more and more objects are found to be simulated with stochastic differential equation,so it has been widely concerned by scholars and has developed rapidly in the last few decades.At the same time,owing the stochastic differential equation is a mathematical model influenced by white noise,so the difficulty of solving it is greatly improved,and the analytical solution is generally unable to be given.Therefore,the numerical methods are needed to deal with it,simulating the original system as much as possible.Numerical methods with good performance have been the goal of many scientists.Besides the high efficiency,these numerical methods should also keep the geometric characteristics of the original system as far as possible,so as to achieve a more realistic exploration of the system.First of all,continuous stage stochastic Runge-Kutta methods for solving the Stratonovich stochastic differential equation are constructed,and mean square convergence of p conditions are given in the form of the root tree.Based on it,the proper coefficient polynomials are selected to obtain the numerical methods of 1 order mean square convergence.The stochastic differential equation of a single integrand is taken into consideration.A random variable is used to construct a class of continuous stage stochastic Runge-Kutta methods with arbitrary high order mean square convergence.Similarly,for the one and two order stochastic differential equations with additional noises,using two random variables to construct the numerical methods of 1.5 mean square order and 2,respectively.Secondly,studying the symplectic properties of the constructed methods for the stochastic Hamilton system and the stochastic Hamilton system with additional noises,giving a class of 1 order and 1.5 order mean square convergence continuous stage stochastic symplectic methods.It is proved that the continuous stage stochastic symplectic Runge-Kutta methods can conserve quadratic invariant of the original system.As the applications of the theory,for each continuous stage stochastic Runge-Kutta method,many classes of numerical methods with the same properties are constructed.