A posteriori error estimates and adaptivity for the IMEX BDF2 method for nonlinear parabolic equations

In this paper, we establish optimal a posteriori error estimates for time discretizations by the IMEX two-step backward differentiation formula (BDF2) method for nonlinear parabolic equations. An effective tool for such derivation is appropriate second-order reconstructions of the piecewise linear approximate solution. We employ the second-order reconstructions to establish the upper and lower error bounds which depend only on the data of the problem and the discretization parameters. By means of the a posteriori error estimates, we design a time adaptive algorithm of IMEX BDF2 method. Numerical experiments for the Allen–Cahn equation with smooth and non-smooth initial data are performed to verify our theoretical results and demonstrate the efficiency of the time adaptive algorithm. In addition, we use the IMEX BDF2 method to solve the Navier–Stokes equations to test the validity of the a posteriori error estimates.