Asymmetry and Condition Number of an Elliptic-Parabolic System for Biological Network Formation

Abstract

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 nonlinear finite difference scheme on a uniform Cartesian grid in a two-dimensional (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 the regularization by the isotropic background permeability $r>0$, showing that the 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 the numerical error analysis of our method making use of the Wasserstein distance.

Keywords

Bionetwork formation / Cai-Hu model / Leaf venation / Asymmetry / Conditioning number / Finite-difference scheme / Semi-implicit / Symmetric alternating direction implicit (ADI) / Wasserstein distance / 35J99 / 35K55 / 35K57 / 65-06 / 65J15 / 76Z99

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