Some Research On Modified SDC Method

The initial value problem of ordinary differential equation(ODE)is a basic def-inite solution problem in the theoretical research and practical application of ODE.Existing analytical methods can only be used to solve some specific types of initial value problems,but the solutions of many practical valuable initial value problems can not be expressed by definite elementary functions,and usually the numerical so-lutions of these problems need to be obtained.The numerical solution is to select a series of discrete points in the solution interval and give the approximate solutions of the initial value problem at these discrete points.At present,the basic methods for numerical solution of initial value problems are:using difference quotient instead of derivative method,numerical integration method,undetermined coefficient method,etc.However,these methods often have a variety of defects,either poor stability,low accuracy,or very large amount of calculation.Therefore,finding numerical al-gorithms with good stability,high accuracy and low computational complexity has always been the goal of Computational Mathematics researchers.Dutt,Greengard and Rokhlin proposed a spectral delay correction(SDC)method for initial value problems of ordinary differential equations in 2000[1].By combin-ing Gauss quadrature with the equivalent Picard integral equation of initial value problems of ordinary differential equations,this method can ac:hieve arbitrary high order accuracy and has good stability.On this basis,the SDC method used in the prediction and correction step of high-order Runge-Kutta(R-K)method has been developed,and obtained that as long as the quadrature nodes are consistent,the convergence rate can be higher.Recently,the author of[2]proposed a modified SDC(modified SDC)method,which extends the high-order convergence behav-ior to non-uniform nodes by improving the regularity of the error function.Based on this modified SDC method,this paper aims to study the influence of integra-tion nodes on the stability and accuracy of the algorithm.The stability images of the algorithm when choosing Chebyshev-Gauss nodes,Chebyshev-Radau nodes and Chebyshev-Lobatto nodes are analyzed in detail,and the computational efficiency of the algorithm when choosing different nodes is compared with numerical examples.