Abstract
The second-order backward-difference (BDF2) method requires solutions at two previous time steps and is often started with the first-order backward-difference (BDF1) method. The initial time step is typically taken to be small in order to minimize the effect of the first-order error. However, contrary to expectations, the solution computed in this way generates a very large error for stiff problems and the error grows as the initial time step is further reduced. As shown in this note, the problem is originated from the fact that the variable-coefficient BDF2 method combined with any first- or higher-order one-step method is asymptotically equivalent to the trapezoidal method and therefore not L-stable. Based on the study presented, it is strongly recommended that BDF2 method be started with a self-starting second-order implicit Runge-Kutta method, which guarantees second-order accuracy and L-stability at no additional cost.
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