ANNOUNCEMENT ON 'SHARP ERROR ESTIMATE OF BDF2 SCHEME WITH VARIABLE TIME STEPS FOR LINEAR REACTION-DIFFUSION EQUATIONS'

In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps. Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time steps τ1=O(τ2) for the first level solution u1. This is, the first-order consistence of the first level solution u1 like BDF1 (i.e. Euler scheme) as a starting point does not cause the loss of global temporal accuracy, and the ratios are updated to rk ≤ 4.8645.