线性反应扩散方程的时间变步长BDF2格式的最优误差估计

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

虽然时间变步长的两步向后差分公式(BDF2)在模拟多尺度动力学具有重要的价值和广泛的应用,但其稳定性和收敛性分析仍不完整.在本工作中,我们重新讨论了线性反应扩散问题的BDF2格式.利用[11]中离散正交卷积(DOC)核的技巧,引入离散互补卷积(DCC)核的概念,我们证明了在相邻时间步长比条件0<rk:=τk/τk−1≤rmax≈4.8645下,BDF2格式是无条件稳定的且具有二阶收敛率.我们的分析表明,二阶收敛性是最优且鲁棒的.鲁棒性指对于任意满足0<rk:=τk/τk−1≤rmax≈4.8645的时间步长,BDF2格式仍保持二阶收敛性,并不需要额外的时间步长比限制条件.此外,我们的分析还表明,当0<rk≤4.8645时,用BDF1(即Euler格式)计算第一步数值解u 1不会导致全局二阶精度的损失.最后,我们给出了数值例子来佐证本文理论分析.

While the variable time-steps two-step backward differentiation formula(BDF2)is valuable and widely used to capture the multi-scale dynamics of model solutions,the stability and convergence of BDF2 with variable time steps still remain incomplete.In this work,we revisit BDF2 scheme for linear diffusion-reaction problem.By using the technique of the discrete orthogonal convolution(DOC)kernels developed in[11],and introducing the concept of the discrete complementary convolution(DCC)kernels,we present that BDF2 scheme is unconditionally stable under a adjacent time-step ratio condition:0<rk:=τk/τk-1≤rmax≈4.8645.With the uses of DOC and DCC kernels,the second-order temporal convergence can be achieved under0<rk≤rmax≈4.8645.Our analysis shows that the second-order convergence is sharp and robust.The robustness means that the second-order convergence is sharp for any time step satisfying0<rk≤rmax≈4.8645,this is,it does not need extra restricted conditions on the time steps.In addition,our analysis also shows that the first level solution u1obtained by BDF1(i.e.Euler scheme)does not cause the loss of global accuracy of second order with 0<rk≤4.8645.Numerical examples are provided to demonstrate our theoretical analysis.