Time adaptive conservative finite volume method
Time adaptive conservative finite volume method
复制标题
时间自适应保守有限体积法
DOI：
10.1016/j.jcp.2019.109067
复制
发表时间：
2020
影响因子：
4.1
通讯作者：
P. Jenny
中科院分区：
文献类型：
--
作者：
P. Jenny
时间相关问题的有限体积方法通常采用恒定时间步长的时间积分方案。后者，为了确保稳定性和时间精确的解决方案，必须选择足够小，使Courant-Friedrichs-Lewy （CFL）数保持在一个临界值以下。然而，在许多情况下，这种时间步长限制导致大量非常小的时间步长，这使得模拟非常昂贵。由于传统时间步进算法的缺点，人们提出了各种子时间步进算法。然而，它们中的大多数本质上是异步的，需要较小的本地CFL数，或者不是严格保守的。本文提出了一种新的有限体积法自适应时间积分方案，该方案具有保守性、高时空阶数、鲁棒性和易于实现的特点。它依赖于局部子时间步长，这些子时间步长是全局时间步长的2次方，即网格单元按其终止时间的顺序异步进行，但由于它们在每个全局时间步长结束时都是同步的，因此可以保证平均通量的连续性，从而在全局时间步长分辨率下严格守恒。一维和二维的数值实验表明，自适应保守时间积分（ACTI）方案在保持较高时空精度的前提下，比传统的时间积分方案具有显著的加速因子。
Finite volume methods for time dependent problems typically employ time integration schemes with constant time step sizes. The latter, in order to ensure stability and time accurate solutions, have to be chosen small enough, such that the Courant-Friedrichs-Lewy (CFL) number stays below a critical value everywhere. In many cases, however, this time step size limitation leads to huge numbers of very small time steps, which renders simulations very expensive. Motivated by this drawback of conventional time stepping schemes, various sub-time stepping algorithms have been proposed. However, most of them are inherently asynchronous, require small local CFL numbers or are not strictly conservative. In this paper a new adaptive time integration scheme for finite volume methods, which is conservative, of high spatial and temporal order, robust and easy to implement is presented. It relies on local sub-time steps which are fractions of a global time step by powers of two, i.e., the grid cells proceed asynchronously in the order of their termination time, but since they all synchronize at the end of each global time step, it is possible to guarantee continuity of the mean fluxes and thus strict conservation at the global time step resolution. Numerical experiments with 1D and 2D test cases demonstrate that the adaptive conservative time integration (ACTI) scheme can achieve extreme speed-up factors over conventional time integration, while still maintaining high spatial and temporal accuracy.