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 utilise 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 splitted 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 method is provided. Several numerical tests are performed to show the efficiency and accuracy of the proposed approach.

翻译：本文提出了一种高效求解非线性高对比度多尺度扩散问题的方法。我们将显式-隐式-零（Explicit-Implicit-Null，EIN）方法融入求解框架，将非线性项分解为线性项与阻尼项两部分，并分别采用隐式和显式时间推进格式进行处理。针对线性部分的多尺度特性，我们进一步引入时间部分显式分裂格式，并构建合适的的多尺度子空间以加速计算。将近似解分裂至这些表征不同物理过程的的子空间中。该时间分裂格式对表示高对比度性质的低维子空间采用隐式离散化，而对其他子空间使用显式离散化。我们分析了所提格式的稳定性，给出了线性扩散系数的选取条件，并证明了该方法的收敛性。最后通过数值算例验证了所提方法的有效性与精度。