论文标题
具有多个尺度的微分方程的错误估计和适应性
Error estimation and adaptivity for differential equations with multiple scales in time
论文作者
论文摘要
我们考虑具有多个量表的普通微分方程系统。通常，我们对慢速变量的长期视野感兴趣，该变量与快速尺度起作用的解决方案组件耦合。尽管快速尺度变量对于耦合问题的动态至关重要，但它们通常对自己毫无兴趣。最近，我们提出了一种时间多尺度方法，该方法适合于异质多尺度方法的框架，并允许具有大量加速的有效仿真。快速和缓慢的尺度是通过引入本地平均值并通过局部定期及时问题代替快速尺度贡献来解耦的。在这里，我们将这种多尺度方法推广到更大类别的问题，但尤其是，我们根据双重加权残差方法得出了A后验误差估计器，该方法允许将误差分解为平均误差，慢速误差和快速尺度上的误差。我们证明了误差估计器的准确性及其用于自适应控制数值多尺度方案的准确性。
We consider systems of ordinary differential equations with multiple scales in time. In general, we are interested in the long time horizon of a slow variable that is coupled to solution components that act on a fast scale. Although the fast scale variables are essential for the dynamics of the coupled problem, they are often of no interest in themselves. Recently we have proposed a temporal multiscale approach that fits into the framework of the heterogeneous multiscale method and that allows for efficient simulations with significant speedups. Fast and slow scales are decoupled by introducing local averages and by replacing fast scale contributions by localized periodic-in-time problems. Here, we generalize this multiscale approach to a larger class of problems but in particular, we derive an a posteriori error estimator based on the dual weighted residual method that allows for a splitting of the error into averaging error, error on the slow scale and error on the fast scale. We demonstrate the accuracy of the error estimator and also its use for adaptive control of a numerical multiscale scheme.
