论文标题
扩散方程的三阶变量bdf时间步长的离散能量分析
Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations
论文作者
论文摘要
这是我们的系列作品之一，用于对可变步骤BDF方案的离散能量分析。在这一部分中，我们介绍了具有线性扩散方程的可变步骤的三阶BDF（BDF3）方案的稳定性和收敛分析，请参见例如[Siam J. Numer。肛门，58：2294-2314]和[数学。 Comp。，90：1207-1226]对于我们以前在BDF2方案上的作品。为此，我们首先在相邻的步骤比小于1.4877的条件下建立可变step BDF3公式的离散梯度结构，通过该公式我们可以建立离散的能量耗散定律。然后获得$ l^2 $规范中的网状稳定稳定性和收敛分析。在这里，网格鲁棒性意味着解决方案误差由最大时步大小很好地控制，但独立于相邻的时步比率。我们还提出了数值测试以支持我们的理论结果。
This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see e.g. [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the $L^2$ norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.
