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.
翻译:这是我们对可变步骤BDF计划进行离散能源分析的系列工作之一。 在这一部分,我们提出对三阶BDF(BDF3)计划(BDF3)的稳定性和趋同性分析,并附有线性扩散方程式的可变步骤,例如,见[SIAM J.Numer.Anal., 58:2294-2314]和[Math.Comp., 90:1207-126],用于我们以前在BDF2计划中的工程。为此,我们首先在相邻步骤BDF3公式中建立一个离散梯度结构,条件是相邻步骤比率小于1.4877,据此我们可以制定离散能源消散法。Mesh-robust稳定性和趋同性分析,然后在$L__2美元的规范中实现。这里的网状坚固度意味着,解决方案错误在最大时限范围内得到很好的控制,但与相邻时间步比率无关。我们还提出数字测试,以支持我们的理论结果。