An Implicit–Explicit Second-Order BDF Numerical Scheme with Variable Steps for Gradient Flows

Abstract

In this paper, we propose and analyze an efficient implicit-explicit (IMEX) secondorder backward differentiation formulation (BDF2) scheme with variable time steps for gradient flow problems using a scalar auxiliary variable (SAV) approach. Comparing with the traditional second-order SAV approach [Shen et al., J. Comput. Phys. 2018], we only use a first-order method to approximate the auxiliary variable. This treatment does not affect the second-order accuracy of the unknown function ϕ, and is essentially important for deriving the unconditional energy stability of the proposed BDF2 scheme with variable time steps. We prove the unconditional energy stability of the scheme for a modified discrete energy with the adjacent time step ratio γn+1 := τn+1/τn ≤ 4.8645. The uniform H 2 bound for the numerical solution is derived under a mild regularity restriction on the initial condition, that is ϕ(x, 0) ∈ H 2 . Based on this uniform bound of the numerical solution, a rigorous error estimate of the proposed scheme is carried out on the nonuniform temporal mesh. Finally, serval numerical tests are provided to validate the theoretical claims. With the application of an adaptive time-stepping strategy, the efficiency of our proposed scheme can be clearly observed in the coarsening dynamics simulation.

A linear adaptive second‐order backward differentiation formulation scheme for the phase field crystal equation

“…To construct a second‐order BDF scheme for (6), we firstly introduce a second order approximation $$$ {F}_2^{j+\sigma}\phi $$$ to $$$ {\phi}^{\prime }(t) $$$ at $$$ t={t}_{j+\sigma } $$$ as follows, seeing also [13, 15]. Here, $$$ {t}_{j+\sigma}:= {t}_n+\sigma {\tau}_{n+1} $$$ with $$$ \frac{1}{2}\le \sigma \le 1.$…”

“…Before rigorously proving the energy stability of the scheme (13), we need the following essential inequality, seeing also [13, 15], $$$$ {\delta}_t{\phi}^{n+1}\cdotp {F}_2^{n+\sigma}\phi \ge \left[g\left({\gamma}_{n+2}\right)+\frac{1}{2}G\left({\gamma}_{n+1},{\gamma}_{n+2}\right)\right]\frac{{\left|{\delta}_t{\phi}^{n+1}\right|}^2}{\tau_{n+1}}-g\left({\gamma}_{n+1}\right)\frac{{\left|{\delta}_t{\phi}^n\right|}^2}{\tau_n}, $$$$ where $$$ {\delta}_t{\phi}^{n+1}={\phi}^{n+1}-{\phi}^n $$$. And in (14), for any $$$ 0<s,z\le {\gamma}_{\ast \ast}\left(\sigma \right) $$$, …”

“…To construct a second-order BDF scheme for (6), we firstly introduce a second order approximation F j+𝜎 2 𝜙 to 𝜙 ′ (t) at t = t j+𝜎 as follows, seeing also [13,15]. Here, t j+𝜎 ∶= t n + 𝜎𝜏 n+1 with 1 2 ≤ 𝜎 ≤ 1.…”

“…However, these adaptive time‐stepping schemes in [21, 33] are also nonlinear and the nonlinear iteration is unavoidable at each time step. Very recently, combining the nonuniform BDF2 formula and the SAV approach, we successfully derived a linear second order unconditionally energy stable time stepping scheme for gradient flows with the adjacent time‐step ratio $$$ {\gamma}_n\le 4.8465 $$$ in [13]. Though it avoids solving the nonlinear system for the phase function $$$ {\phi}^n $$$ by explicit treatment for the nonlinear term, there still exits a nonlinear algebraic equation to be solved for the auxiliary variable at each time step.…”

“…The first order treatment of the auxiliary variable will not affect the second order accuracy of the unknown atom density field $$$ {\phi}^n $$$ by setting the positive constant $$$ {C}_0 $$$ large enough such that $$$ {C}_0\ge 1/\tau $$$. Moreover, the constructed adaptive BDF2 scheme avoids solving the nonlinear algebraic equation for the auxiliary variable as needed in our previous work [13]. More precisely, there are only two sixth‐order equations with constant coefficients to be solved at each time step, which can be solved efficiently by exiting fast Poisson solvers.…”