Some Research On Parareal Algorithm

The Parareal algorithm is an iterative method of parallelization in the time direction.it divides the entire time period into several subintervals,and configures a processor for each subinterval,and then executes the task simultaneously.This method can reduce the calculation time as much as possible while maintaining high accuracy.The goal of this thesis is to find a Parareal algorithm for some combination of coarse and fine propagators,which possesses high accuracy and stability when the number of iterations is small.This thesis uses the stability function of the Parareal algorithm to get some theoretical conclusions about stability.Then the influence of iteration times on the stability of Parareal algorithm is studied,the Parareal algorithms with different com-binations of coarse cand fine propagators are compared,as the number of iterations increases,their stable region changes along the negative semi-real axis.Finally,in the numerical experiment section,the relationship between the number of iterations and the error in the end point is studied.Since the spectral deferred correction method(abbreviated as the SDC method)can achieve arbitrarily high order accuracy and possess good stability.So we use SDC method as the fine propagator,and find that when the coarse propagator uses the implicit euler method,it can maintain high accuracy and stability when the number of iterations is less.