We compare the solutions of two systems of partial differential equations (PDE), seen as two different interpretations of the same model that describes formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic-parabolic PDE system for the conductivity vector $m$, the conductivity tensor $\mathbb{C}$ and the pressure $p$. We use finite differences schemes in a uniform Cartesian grid in the spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved by a semi-implicit scheme in time. Since the conductivity vector and tensor appear also in the Poisson equation for the pressure $p$, the elliptic equation depends implicitly on time. For this reason we compute the solution of three linear systems in the case of the conductivity vector $m\in\mathbb{R}^2$, and four linear systems in the case of the symmetric conductivity tensor $\mathbb{C}\in\mathbb{R}^{2\times 2}$, at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved, to see the differences in the solutions of the two systems.
翻译:我们比较了两种局部差异方程式(PDE)的解决方案,认为它是描述复杂生物网络形成模式的同一模型的两种不同的解释。两种方法都考虑到介质通过网络流动的介质的时间演变情况。我们计算了导电矢量的椭圆parPDE系统的解决方案($m美元)、导电量 $\mathbb{C}美元和压力美元。我们在空间二维环境中使用一个统一的碳酸盐网格中的有限差异方案来解决两种系统,两种系统都是通过一个半隐含的办法及时解决的。两种方法都考虑到通过网络流动的介质的时间演变情况。由于导量矢量矢量矢量的矢量和弧值也出现在Poisson方程式的压力 $p$,因此我们计算了三种线性系统的解决方案的解决方案($m\ mathb{R ⁇ 2} 和四个线性系统,我们通过一个半隐含隐含的公式来解决这两个系统。 加速了方向2号的计算方法。