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

• Published in: BIT Vol. 62; no. 4; pp. 1789 - 1822
• Main Authors: Bourgault, Yves, Garon, André
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2022; Springer Nature B.V
• Subjects: Accuracy; Boundary value problems; Computational Mathematics and Numerical Analysis; Differential equations; Differentiation; Formulas (mathematics); Mathematics; Mathematics and Statistics; Numeric Computing; Ordinary differential equations; Stability; 65L05; Backward differentiation formulae; High-order time-stepping methods; 65L04; Ordinary differential equations; 65L20; 5B05; 65L12; Deferred correction; A-stability
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-022-00931-1

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 (DC p ) results in a p + 1 -order accurate solution (DC p + 1 ). 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 DC p to DC p + 2 . These requirements include the constant time step DC p methods. We also prove that all these DC p methods are A-stable. We briefly discuss two other DC variants to illustrate how a proper transition from DC p to DC p + 1 is critical to maintaining A-stability at all orders. Numerical experiments based on two manufactured (closed-form) solutions confirmed the accuracy orders of the DC p – for DC p , p = 2 , 3 , 4 , 5 – 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 DC p solution with the DC p + 1 solution, and a last test case that our new methods maintain their order of accuracy for a stiff system.