The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring the approximate solution's stability. In this paper, we construct unconditionally stable first- and second-order accuracy schemes on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.

翻译：针对二阶演化方程柯西问题的近似求解，首先采用三层时间近似方法。当使用均匀时间网格时，此类近似易于构造且研究相对简单。在数值求解应用问题时，应关注变时间步长的近似方法。当在非均匀网格上使用多层格式时，需通过选择适当的近似方法并确保近似解的稳定性来维持精度。本文针对二阶演化方程的柯西问题近似求解，在非均匀时间网格上构造了无条件稳定的一阶和二阶精度格式。我们采用特殊变换将原始二阶微分-算子方程转化为一阶方程组。对一阶方程组应用标准的两层时间近似方法，获得了有限维希尔伯特空间中关于初始数据和右端项的稳定性估计。消去辅助变量即可得到原始二阶演化方程的三层格式。针对空间一维双抛物型方程的测试问题开展了数值实验，在网格步长随机变化的非均匀网格上验证了所构造格式的精度和稳定性特性。