Second derivative numerical differentiation formulas for stiff systems of ODEs

The SDBDF methods, which extend BDF methods by incorporating the second derivative of the solution, have attracted interest due to their favorable convergence and stability properties. Inspired by the approach used in developing numerical differentiation formulas for solving stiff ODEs—those employed in Matlab ’s ode15s solver—this paper proposes a modification to the SDBDF methods that reduces the local truncation error, with only a slight loss of stability in higher-order cases. The analysis of the newly designed methods demonstrates their capability and efficiency in solving stiff initial value problems for ordinary differential equations. Numerical experiments on several well-known stiff problems confirm the effectiveness of the proposed methods.