We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.
翻译:我们提出了一般连续性方程式的有限量计划,这些方程式保留了流动功能饱和效应所产生的正负性和全球界限;对于梯度流,这些方程式分散了完全独立的自由能源;此外,这些方程式被概括为非线性连续性方程式的结合系统,如数学物理学或生物学的多物种模型、保护界限和酌情消除能量的多物种模型;这些结果通过广泛的数字模拟加以说明,这些模拟探索生物学中已知的行为,并展示文献尚未描述的新现象。