Research On Weighted Adams Method And Step-Size Control

The mathematical models for many stiff systems,such as chemical reaction processes,the operation of automatic control systems,are initial value problems for stiff differential equation.The simulation of stiff system is essentially the process of solving stiff differential equation.The traditional numerical algorithm with fixed step-size can not take into account both efficiency and accuracy.A single step-size controller cannot satisfy a time-varying system.Therefore,it is necessary to design an adaptive step-size method that can improve computing efficiency.In order to improve the efficiency of solving stiff differential equations,an Adaptive Step-size Weighted Adams method is proposed based on the feedback control theory.This algorithm can reduce the total number of simulation steps and the average number of Newton iterations of each step,thus reducing the calculation time and improving the efficiency of solving stiff differential equations.It has application value in the numerical simulation of stiff dynamic systems such as robots and aircraft.The main research work of this paper is as follows:Firstly,this paper analyzes the influence of the eigenvalues of Jacobian matrix on the selection of stepsize and proposes a modified Newton method based on Newton iteration method to accelerate the convergence of Newton iteration.Inspired by the iterative format of the modified Newton method for solving nonlinear equations,a weighted Adams method with similar iterative format is designed.The method combines the explicit and the implicit Adams method of the same order by weighting.The modified Newton iteration format for the stiff differential equation using the weighted Adams method is consistent with the modified Newton iteration format for the stiff differential equation using only the implicit Adams method,which can ensure the stability of the algorithm.Secondly,in order to realize the adaptive adjustment of step-size,this paper designs a closed-loop feedback system according to the basic control theory so that can select the optimal step-size of the next step according to the local error of each step.This paper also analyzes the relationship of the eigenvalue,stiffness ratio and the selection of optimal step-size controller.It is found that the relation among stiffness ratio,eigenvalue,and optimal step controller is not linear,which cannot be expressed by a definite formula.In order to describe this relationship,this paper uses BP neural network for fitting,uses experimental equations to collect data sets,and finally uses the data set for neural network training.The training results show that the network has high accuracy and can be used to select the step controller in the process of solving the stiff differential equation.Finally,the weighted Adams method and the step controller based on BP neural network are combined to form a numerical algorithm which can realize the adaptive variable step-size.The algorithm can select the optimal step-size controller in real time according to the eigenvalue of Jacobian matrix and stiffness ratio of system by using neural network.In this paper,the method is applied to the numerical simulation of two stiff systems.The experimental simulation results show that compared with the classical implicit Adams method and BDF method,Adaptive Step-size Weighted Adams method proposed in this paper can significantly reduce the total number of solving steps and the average number of Newton iterations per step on the premise of ensuring the accuracy,which greatly improves the solving efficiency.