This work continues a line of works on developing partially explicit methods for multiscale problems. In our previous works, we have considered linear multiscale problems, where the spatial heterogeneities are at subgrid level and are not resolved. In these works, we have introduced contrast-independent partially explicit time discretizations for linear equations. The contrast-independent partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting is proposed that treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that it guarantees a stability and formulate a condition for the time stepping. This condition is formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.

翻译：这项工作在为多尺度问题开发部分清晰的方法方面继续展开工作。 在我们以前的作品中, 我们考虑了线性多尺度问题, 空间异质处于亚格罗水平, 并且没有得到解决。 在这些作品中, 我们引入了线性方程式的对比独立部分明确的时间分解。 对比独立部分时间分解将空间空间分为两个部分: 对比依赖( 快速) 和对比独立( 低) 空间, 由多尺度空间分解定义。 在进行这种分解后, 提议时间分解, 暗含地处理快速组件, 并明确处理慢速组件。 空间分解和时间分解是选择的, 以保障稳定, 并为时间间隔设定一个条件。 在本文中, 我们将此方法扩展为非线性问题。 我们提出分解方法, 并得出一个保证稳定性的条件 。 这个条件要求某种差异独立空间的空域, 以缓慢的方块组成。 我们提出数字结果, 并表明拟议的方法提供了类似隐含方法的结果, 且时间步骤独立于对比之外的时间步骤 。