确定性粒子方法对接触抑制交叉扩散问题的近似
Deterministic particle method approximation of a contact inhibition cross-diffusion problem
AI总结：
本文研究了利用确定性粒子方法对两种种群的演化交叉扩散问题进行数值近似，探讨了交叉扩散问题在不同扩散参数下的分类，并比较了有限元方法与粒子方法在接触抑制问题中的数值实验结果。
摘要：
我们使用确定性粒子方法来生成演化交叉扩散问题两个种群解的数值近似。根据与内种群和外种群排斥强度相关的扩散参数值，系统可以根据关联矩阵进行分类。当矩阵为正定矩阵时，问题有良好定义，有限元近似能产生收敛于精确解的近似解。特别重要的是当矩阵仅为半正定且初始数据分离时，即接触抑制问题。在这种情况下，解可能是不连续的，因此（符合）有限元近似可能在不连续附近表现出不稳定性。本文推导了粒子方法对一般交叉扩散问题的近似，并将其应用于接触抑制问题。然后我们提供了一些数值实验，比较有限元方法和粒子方法离散化所产生的结果。
英文摘要：
We use a deterministic particle method to produce numerical approximations to the solutions of an evolution cross-diffusion problem for two populations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. When the matrix is definite positive, the problem is well posed and the Finite Element approximation produces convergent approximations to the exact solution. A particularly important case arises when the matrix is only positive semi-definite and the initial data are segregated: the contact inhibition problem. In this case, the solutions may be discontinuous and hence the (conforming) Finite Element approximation may exhibit instabilities in the neighborhood of the discontinuity. In this article we deduce the particle method approximation to the general cross-diffusion problem and apply it to the contact inhibition problem. We then provide some numerical experiments comparing the results produced by the Finite Element and the particle method discretizations.
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