Theoretical Study On Stability Of Explicit Spectral Deferred Correction Method

There are two deficiencies in use of the classic deferred correction(abbreviated as DC)method to solve differential equations in numerical methods,They are the insta-bility of the numerical differential process and the Runge phenomenon of equidistant nodes.Spectral deferred correction(abbreviated as SDC)Method to eliminate these deficiencies by using integral matrices and selecting spectral points.SDC method can achieve any high-order accuracy and has good stability.At present,the existing literatures have calculated the stable region by numerical methods,but there is no theoretical calculation of the exact expression of the amplification factor.This paper aims to study the stability of explicit SDC methods.Specifically,we get a exact expression of the amplification factor through theoretical calcula-tion,and then draw the stability regions.This paper selects equispaced nodes and Chebyshev-Guass-Lobatto orthogonal nodes for discussion,and considers the fol-lowing three cases respectively.First,the exact expression of the amplification factor of the DC method based on the forward Euler method when 4 equispaced nodes are selected.Secondly,the exact expression of the amplification factor of the SDC method based on the forward Euler method when 4 equispaced nodes and 4 CGL orthogonal nodes are selected respectively.Finally,the exact expression of the am-plification factor of the SDC method based on the 2nd order explicit Runge-Kutta method is selected when 4 equidistant nodes and 4 CGL orthogonal nodes axe se-lected respectively.By comparison,it is found that in the case of 4 equispaced nodes,the SDC method based on forward Euler method has a larger stability region than the corresponding DC method,The SDC method using CGL orthogonal nodes has a larger stability region than equispaced nodes.Although this conclusion has many experimental results,the theoretical analysis in this article is original.