Variable-step deferred correction methods based on backward differentiation formulae for ordinary differential equations

This paper presents a sequence of variable time step deferred correction (DC) methods constructed recursively from the second-order backward differentiation formula (BDF2) applied to the numerical solution of initial value problems for first-order ordinary differential equations (ODE). The sequence of corrections starts with the BDF2 then considered as DC2. We prove that this improvement from a p-order solution (DCp) results in a p+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p+1$$\end{document}-order accurate solution (DCp+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p+1$$\end{document}). This one-order increment in accuracy holds for the least stringent BDF2 0-stability conditions. If we introduce additional requirements for the ratio of consecutive variable time step sizes, then the order increment is 2, allowing a direct transition from DCp to DCp+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p+2$$\end{document}. These requirements include the constant time step DCp methods. We also prove that all these DCp methods are A-stable. We briefly discuss two other DC variants to illustrate how a proper transition from DCp to DCp+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p+1$$\end{document} is critical to maintaining A-stability at all orders. Numerical experiments based on two manufactured (closed-form) solutions confirmed the accuracy orders of the DCp – for DCp, p=2,3,4,5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2,3,4,5$$\end{document} – both with constant or alternating time step sizes. We showed that the theoretical conditions required to obtain an increment of orders 1 and 2 are satisfied in practice. Finally, a test case shows that we can estimate the error on the DCp solution with the DCp+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p+1$$\end{document} solution, and a last test case that our new methods maintain their order of accuracy for a stiff system.