# Second order predictor-corrector pairs

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Carol Marley Franklin (Creator)
Institution
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
Michael Willett

Abstract: The purpose of this study is to introduce multistep methods for approximating the solutions of ordinary differential equations and to study the general convergent, second order, predictor-corrector pair. Given the values of a solution function, either exact or approximate, at certain points on the x-axis, a multistep method approximates the values of that function at other points on the x-axis. Necessary and sufficient conditions for the convergence of a multistep method are the conditions of consistency and stability. A multistep method which approximates the value of the solution function y at xN with no knowledge of the value of the derivative y1 at xN is called a predictor. A corrector is a multistep method which uses the value of the derivative of y at xN to approximate y at xN. The predictor is used to generate a rough estimate of the solution function at a point x then the corrector is used to correct the predicted value at xN . Used together they constitute a predictor-corrector pair. If the differential equation to be solved is of the form y' = ?y, the predictor can be substituted in the corrector to yield a linear recursion. Constraints on the roots of the recursion which would ensure convergence to the true solution are discussed along with their applicability to equations of the form y' = f(x,y).