非线性平流扩散方程的非重叠Schwarz波形松弛

M. Gander, S. Lunowa, C. Rohde
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引用次数: 4

摘要

非线性平流扩散方程经常出现在输运过程的模拟中。对于这些方程,我们提出了一种Schwarz波形松弛型的无重叠区域分解算法。它依赖于子域之间的非线性零阶(或Robin)传输条件,以确保收敛解及其在界面上的法向通量的连续性。证明了该算法的唯一迭代解的存在性和收敛性。然后,我们采用时间上的正演欧拉离散法和空间上的有限体积法,给出了求解SWR问题的数值离散化方法,包括求解非线性传输条件的局部牛顿迭代。我们的离散算法是渐近保持的,即在消失的粘度极限下是鲁棒的。最后,我们给出的数值结果证实了理论发现,特别是算法的收敛性。此外,我们还证明了SWR算法可以成功地应用于多孔介质中的两相流问题,作为具有强非线性平流和扩散通量的演化方程的范例。
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Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations
Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e. robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes.
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