通过强耦合双尺度系统对非线性色散效应的数值探索
Numerical Exploration of Nonlinear Dispersion Effects via a Strongly Coupled Two-scale System
AI总结：
研究物质在多孔介质中传输的非线性色散效应，通过强耦合双尺度系统建模。引入皮卡型迭代和时间离散解耦两种数值格式，并提出预计算策略，有效减少计算时间且收敛到精确解，通过实验展示了相关色散效应。
摘要：
物质在多孔介质中的有效快速传输常具有复杂的色散效应。为用数学描述此类情况，需考虑尺度间完全耦合的双尺度边值问题，宏观传输非线性依赖局部漂移相互作用且受局部浓度影响。此类双尺度问题计算昂贵，因众多椭圆型偏微分方程需不断重新计算。本文研究含适当非线性色散项的有效双尺度模型并数值探索其弱解行为。引入处理相同非线性尺度耦合的两种不同数值格式：皮卡型迭代和时间离散解耦。还提出预计算策略，将单元问题计算推至离线阶段。该方法对两种格式均有效，显著减少计算时间且收敛到精确解。最后通过数值实验测试格式，展示特定微观结构和模型成分选择引入的色散效应。
英文摘要：
The effective, fast transport of matter through porous media is often characterized by complex dispersion effects. To describe in mathematical terms such situations, instead of a simple macroscopic equation (as in the classical Darcy's law), one may need to consider two-scale boundary-value problems with full coupling between the scales where the macroscopic transport depends non-linearly on local (i.e. microscopic) drift interactions, which are again influenced by local concentrations. Such two-scale problems are computationally very expensive as numerous elliptic partial differential equations (cell problems) have to constantly be recomputed. In this work, we investigate such an effective two-scale model involving a suitable nonlinear dispersion term and explore numerically the behavior of its weak solutions. We introduce two distinct numerical schemes dealing with the same non-linear scale-coupling: (i) a Picard-type iteration and (ii) a time discretization decoupling. In addition, we propose a precomputing strategy where the calculations of cell problems are pushed into an offline phase. Our approach works for both schemes and significantly reduces computation times. We prove that the proposed precomputing strategy converges to the exact solution. Finally, we test our schemes via several numerical experiments that illustrate dispersion effects introduced by specific choices of microstructure and model ingredients.
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