Comparison of implicit-explicit and Newton linearized variable two-step BDF methods for semilinear parabolic equations

It is interesting to compare implicit-explicit (IMEX) and Newton linearized (NL) methods since they are two classes of typical time discretization methods for solving nonlinear differential equations. In this paper, we compare IMEX with NL two-step backward differentiation formula (BDF2) methods with variable step-size for solving semilinear parabolic differential equations. Under the appropriate time-step ratio restriction, the stability of the two methods is established by energy estimates and recent novel technique. Based on these stability results, the a priori error bounds for these methods are also derived. Numerical results not only illustrate the feasibility of the proposed method for solving semilinear parabolic differential equations but also reveal that IMEX BDF2 method is more effective than NL BDF2 method.