We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation. The numerical method is based on a non-linear finite difference scheme on a uniform Cartesian grid in a 2D domain. The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution. In particular, we show that using the symmetric alternating-direction implicit (ADI) method for time discretization helps preserve the symmetry of the solution, compared to the (nonsymmetric) ADI method. Moreover, we study the effect of regularization by isotropic background permeability $r>0$, showing that increased condition number of the elliptic problem due to decreasing value of $r$ leads to loss of symmetry. Finally, we perform numerical error analysis of our method in Wasserstein distance.

翻译：我们展示了张量值椭圆-抛物型偏微分方程模型在生物网络形成中的数值模拟结果。该数值方法基于二维区域均匀笛卡尔网格上的非线性有限差分格式。重点研究了不同离散化方法和正则化参数选择对数值解对称性的影响。具体而言，我们证明与（非对称）交替方向隐式（ADI）方法相比，采用对称交替方向隐式（ADI）方法进行时间离散化有助于保持解的对称性。此外，我们研究了各向同性背景渗透率$r>0$正则化效应，表明由于$r$值减小导致椭圆问题条件数增加会引发对称性丧失。最后，我们在Wasserstein距离下对所提方法进行了数值误差分析。