We present 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 stiff first-order ordinary differential equations, such as those resulting from the spatial dicretization of partial differential equations. The sequence of

corrections starts with BDF2 then considered as DC2. In our talk, we will explain how

critical is the ratio of consecutive time steps $\omega_n := \frac{k_n}{k_{n-1}}$,

not only to guarantee stability but also to obtain a maximal improvement of order

of accuracy in DC methods. In general, the improvement from a $p$-order solution (DCp)

results in a $p+1$-order accurate solution (DCp+1). This one-order increment in

accuracy holds for the least stringent BDF2 0-stability conditions, i.e., $\omega_n

\le \omega < 1+\sqrt{2}$. If we introduce additional requirements for the

ratio of consecutive variable time step sizes, i.e., $|\omega_n-\omega_{n-1}|\le\mathcal{C}k$, then the order increment is 2, allowing a direct transition from DCp to DCp+2. These requirements include the

constant time step DCp methods. Interestingly, 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 is critical to maintain A-stability at all orders. Numerical results

for ODEs and PDEs will be presented. For instance, numerical experiments based on

manufactured (closed-form) solutions confirmed the accuracy orders of the DCp – for DCp, $p=2,3,4,5$ -- 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 very accurately estimate the error on the DCp solution with the DCp+1 solution. Beyond more standard error estimators based on the local truncation error, this approach combining DCp-DCp+1 solutions could lead to efficient and reliable adaptive A-stable time-stepping DC methods of arbitrary orders.