论文标题
隐式 - 明确的二阶BDF数值方案，具有可变步骤的梯度流量
An implicit--explicit second order BDF numerical scheme with variable steps for gradient flows
论文作者
论文摘要
在本文中，我们提出并分析了有效的隐式 - 阐释（IMEX）二阶二阶向后分化公式（BDF2）方案，具有使用标量辅助变量（SAV）方法的梯度流量问题的可变时间步骤。我们证明了该方案的无条件能量稳定性，用于修改的离散能量，其相邻的时间步长$γ_{n+1}：= \ dt_ {n+1}/\ dt_ {n} \ leq 4.8645 $。统一的$ h^{2} $用于数值解决方案的限制在初始条件的轻度规则限制下，即$ ϕ（\ x，0）\ in H^{2} $。基于这种均匀的结合，在暂时的非均匀网格上进行了数值溶液的严格误差估计。最后，提供了用于验证理论主张的服务数值测试。通过适用自适应时间稳定策略，在粗糙的动力学模拟中可以清楚地观察到我们所提出的方案的效率。
In this paper, we propose and analyze an efficient implicit--explicit (IMEX) second order in time backward differentiation formulation (BDF2) scheme with variable time steps for gradient flow problems using the scalar auxiliary variable (SAV) approach. We prove the unconditional energy stability of the scheme for a modified discrete energy with the adjacent time step ratio $γ_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$. The uniform $H^{2}$ bound for the numerical solution is derived under a mild regularity restriction on the initial condition, that is $ϕ(\x,0)\in H^{2}$. Based on this uniform bound, a rigorous error estimate of the numerical solution is carried out on the temporal nonuniform 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.
