In this work, we present an efficient approach to solve nonlinear high-contrast multiscale diffusion problems. We incorporate the explicit–implicit-null (EIN) method to separate the nonlinear term into a linear term and a damping term, and then utilize the implicit and explicit time marching scheme for the two parts respectively. Due to the multiscale property of the linear part, we further introduce a temporal partially explicit splitting scheme and construct suitable multiscale subspaces to speed up the computation. The approximated solution is split into these subspaces associated with different physics. The temporal splitting scheme employs implicit discretization in the subspace with small dimension that representing the high-contrast property and uses explicit discretization for the other subspace. We exploit the stability of the proposed scheme and give the condition for the choice of the linear diffusion coefficient. The convergence of the proposed scheme is provided. Several numerical tests are performed to show the efficiency and accuracy of the proposed scheme.

中文翻译：

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非线性多尺度流动问题的显隐零值法的部分显式分裂方案


在这项工作中，我们提出了一种解决非线性高对比度多尺度扩散问题的有效方法。我们采用显式-隐式-零（EIN）方法将非线性项分离为线性项和阻尼项，然后分别对这两部分使用隐式和显式时间推进方案。由于线性部分的多尺度特性，我们进一步引入时间部分显式分割方案并构造合适的多尺度子空间以加速计算。近似解被分成与不同物理场相关的这些子空间。时间分割方案在代表高对比度特性的小维子空间中采用隐式离散化，而对其他子空间采用显式离散化。我们利用所提出方案的稳定性并给出了选择线性扩散系数的条件。提供了所提出方案的收敛性。进行了多次数值测试以显示所提出方案的效率和准确性。