Several Waveform Relaxation Methods For Solving Some Classes Of Stochastic Differential Equations

With the development of modern science and high technology,many mathematical physical models in the fields of industrial engineering,mechanical manufacturing,aerospace,electrical construction and other fields are becoming more and more complex,the scale of those models is also increasing.Many of these mathematical physical models can be described by stochastic differential algebraic systems.General numerical methods have more or less shortcomings in solving these systems,such as long calculation time,untimely response,inability to maintain structures of systems,etc.Therefore,it is of certain application value to construct fast iteration methods for large-scale dynamic systems,and it is particularly important that numerical iteration methods can further maintain essential characteristics of original systems while calculating rapidly.Waveform relaxation methods can decompose a large system into multiple weakly coupled subsystems.Each subsystem can be calculated independently until convergence.Taking this as the starting point,this dissertation studies waveform relaxation methods for solving stochastic differential equations.The main work is as follows:Firstly,for stochastic differential equations driven by 1-dimensional standard Wiener process,split functions are selected and split forms of stochastic differential equations are constructed.Numerical methods with two stochastic variables are used to solve them,corresponding waveform relaxation methods are obtained.It is proved that continuous approximation sequences of waveform relaxation methods converge to certain limit functions by the principle of contraction mapping,the strong first-order convergence conditions of waveform relaxation methods are derived,and a class of Jacobi waveform relaxation methods satisfying order conditions are obtained.In the part of numerical experiments,the convergence order of this kind of waveform relaxation methods is verified,and compared with general numerical methods,it shows that waveform relaxation methods have advantages in rapid iteration.Secondly,high-dimensional multi-noise Ito stochastic differential equations and Stratonovich stochastic differential equations are considered.The convergence order of stochastic continuous Runge-Kutta methods is analyzed by stochastic B-series,and relaxed stochastic differential equations are solved by stochastic continuous Runge-Kutta methods,and then waveform relaxation stochastic continuous Runge-Kutta methods are constructed.Through convergence analysis,it is proved that constructed waveform relaxation stochastic continuous Runge-Kutta methods can converge to high order.In numerical experiments,dimensions of stochastic differential equations are gradually increased,and simulation effects of waveform relaxation stochastic continuous Runge-Kutta methods are observed.Thirdly,stochastic Hamiltonian systems driven by multiplicative noises are considered.By studying wedge products of numerical solutions of waveform relaxation methods,sufficient conditions for waveform relaxation methods to maintain symplectic structures of stochastic Hamiltonian systems are obtained,and two waveform relaxation symplectic methods that can achieve strong first-order convergence are constructed.A waveform relaxation method which does not satisfy symplectic conditions is used to do comparative experiments,and then the importance of symplectic conditions is explained.Finally,stochastic Runge-Kutta methods are directly relaxed to obtain corresponding stochastic Runge-Kutta-Type waveform relaxation methods.The convergence of waveform relaxation methods is analyzed by stochastic Runge-Kutta methods.For stochastic Hamiltonian systems,symplectic-preserving properties of waveform relaxation methods are studied,and two concrete stochastic symplectic Runge-Kutta-Jacobi waveform relaxation methods and stochastic symplectic Runge-Kutta-Gauss-Seidel waveform relaxation methods are constructed.The convergence and symplectic-preserving properties of waveform relaxation methods are verified by numerical experiments.