We introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the associated difficulties is a semi-Lagrangian based reconstruction of subgrid variability into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a comparison to standard methods to support our ideas and discuss possible extensions to other types of Galerkin methods, higher dimensions and nonlinear problems.

我们引入了一种新的数值多尺度方法框架，用于解决由气候科学引发的对流占主导地位的问题。当前的数值多尺度方法（MsFEM）在平稳椭圆问题上效果很好，但是当模型涉及主要的低阶项时会遇到困难。我们克服相关困难的想法是通过解决许多局部逆问题，将基于半拉格朗日的子网格可变性重构为多尺度基础。在全球范围内，该方法看起来像是具有多尺度稳定基础的欧拉方法。我们展示了在一维和二维中运行的示例，并与标准方法进行了比较以支持我们的想法，并讨论了对其他类型的Galerkin方法，更高维和非线性问题的可能扩展。