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 (non symmetric) 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. We show that in this case, neither the use of the symmetric ADI method preserves the symmetry of the solution. Finally, we perform numerical error analysis of our method making use of Wasserstein distance.

翻译：我们呈现了张量化椭圆-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。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。研究重点聚焦于不同离散化方法及正则化参数选择对数值解对称性的影响。特别地，我们证明与（非对称）交替方向隐式法相比，采用对称交替方向隐式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$r>0$的正则化效应，结果表明因$r$值减小导致的椭圆问题条件数增大会引发对称性丧失。我们证明在此情形下，即使使用对称交替方向隐式法也无法维持解的对称性。最终，我们借助Wasserstein距离对方法进行了数值误差分析。