Asymmetry and condition number of an elliptic-parabolic system for biological network formation

TL;DR

This paper investigates how discretization choices and regularization parameters affect the symmetry and condition number in numerical simulations of an elliptic-parabolic PDE model for biological network formation, highlighting the importance of method selection.

Contribution

It demonstrates that symmetric ADI methods help preserve solution symmetry and analyzes how regularization impacts the condition number and symmetry in the model.

Findings

Symmetric ADI preserves symmetry better than non-symmetric ADI.

Increasing regularization parameter r worsens the condition number and reduces symmetry.

Numerical error analysis using Wasserstein distance provides insights into method accuracy.

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 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.