数学杂志2021，Vol.41Issue(6)：471-488,18.
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
摘要
Abstract
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 re-visit 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 dis-crete 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 under 0＜rk≤rmax ≈ 4.8645.Our analysis shows that the second-order convergence is sharp and ro-bust.The robustness means that the second-order convergence is sharp for any time step satisfying 0＜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 u1 obtained 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.关键词
BDF2/DOC/DCC/时间变步长/最优误差估计Key words
BDF2/DOC/DCC/variable time steps/sharp error estimate分类
数理科学引用本文复制引用
张继伟,赵成超..线性反应扩散方程的时间变步长BDF2格式的最优误差估计[J].数学杂志,2021,41(6):471-488,18.基金项目
Supported by NSFC under grant Nos.11771035 and NSAF U1930402,the Natural Science Foundation of Hubei Province No.2019CFA007.The numerical simulations in this work have been done on the supercomputing system in the Supercomputing Center of Wuhan University. （）
