Approximate Preservation Of Quadratic Invariants By Stochastic Partitioned Runge-Kutta Methods

As an extension of the deterministic differential equations,stochastic differential equations(SDEs)are always taking stochastic disturbance into account when describing the change regularity of nature,so more realistic models were established.However,due to the nonlinear property and couple property in SDEs,there are only little special equations that could get exact solutions directly.As a result,it is necessary to make a deep research on numerical methods for SDEs,and designing numerical methods which could preserve the structural characteristics of the original system is an attracting field of research.Stochastic partitioned Runge-Kutta(SPRK)methods preserving quadratic invariants accurately always need to satisfy some certain conditions which indicate that they are fully implicit,it is not easy to implement directly because of the computational complexity.In order to improve computational efficiency,based on the definition of approximate preservation of quadratic invariants for numerical methods,for a stochastic partitioned differential equation containing a 1-dimension standard Wiener process,explicit methods and implicit methods using fixed pointed iteration are discussed respectively which will result in preserving quadratic invariants approximately of the original stochastic system.Explicit SPRK methods,considering the advantage of simple calculation and small time-space complexity et al,the conditions of explicit SPRK methods to preserve quadratic invariants approximately up to certain orders of accuracy are given by means of 4-colored rooted tree and stochastic P-series.These conditions provide an effective approach of constructing this type of numerical methods and thereby constructing a numerical method with order of 2.0-? which acts on the stochastic Kubo oscillator and verifies the effectiveness of the theory.Furthermore,for the implicit SPRK methods to preserve quadratic invariants accurately,fixed-point iteration approach is used.Due to the error of the iteration,it will make a loss of preserving quadratic invariants accurately.In this paper,we give the error bounds of preserving quadratic invariants which is caused by iterative accumulation and supply logical suggestion about the selection of iterative parameter.It is indicated that the error bound caused by the iteration is related to the number of iteration and the time step.At the end of this chapter,a 4-dimension SDEs are used to test the theoretical results.