# Comparison of Two Aspects of a PDE Model for Biological Network Formation

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Numerical Schemes

#### 3.1. Space Discretization

#### 3.2. Time Discretization: Symmetric ADI Method

#### 3.2.1. Time Discretization for the Conductivity Vector

#### 3.2.2. Time Discretization for the Conductivity Tensor

## 4. Numerical Results

#### 4.1. Accuracy Tests and Qualitative Agreements

#### 4.2. Quantitative Agreement

#### Alternative Boundary Conditions

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**In this figure, we show three different quantities of the same computations, with the parameters defined in TestD: the module of the variables at final time (

**left panels**), the flux also at final time (

**central panels**) and the energy as a function of time (

**right panels**). The first row concerns variable m, and the second one is for variable $\mathbb{C}$.

**Figure 2.**In this figure, we show the difference in the results on varying the diffusivity D. In the first column, we have the results of TestG ($D=0.05$); in the second column, we choose the parameters of TestD ($D=0.01$), and in the third one, TestE ($D=0.001$). The first row shows the results of variable m, and the second row shows those of variable $\mathbb{C}$.

**Figure 3.**In this figure, we show the difference in the results on varying the relaxation exponent $\gamma $. In the first column, we have the results of TestH ($\gamma $ = 1); in the second column, we choose the parameters of TestD ($\gamma =0.75$); and in the third one, TestF ($\gamma =0.5$). The first row shows the results of variable m, and the second row, variable $\mathbb{C}$.

**Figure 4.**In this figure, we show the results for variable $\mathbb{C}$ varying the parameter $\epsilon $. On the left, we have the results of TestI ($\epsilon ={10}^{-2}$), in the central panel, TestD$(\epsilon ={10}^{-3})$, and on the right, TestL$(\epsilon ={10}^{-4})$.

**Figure 5.**Comparison between TestN and TestO, with initial condition defined in Equation (29). The main difference is that on the left, we have $D=0$, while on the right, we have $D={10}^{-5}$.

**Figure 6.**Comparison between TestN and TestO. Here, the initial condition is defined in Equation (41), with $D=0$ on the left and $D={10}^{-5}$ on the right.

**Figure 7.**In this figure, we show the difference between two different solutions choosing, as initial conditions, ${m}^{0,1},{m}^{0,2}$ (

**left**) and ${m}^{0,1},{m}^{0,3}$ (

**right**). We plot the expression defined in Equation (49) as a function of time, with $\gamma =1.75>1$, and the others parameters are defined in TestD.

**Figure 8.**In this figure, we show the steady states when $\gamma =0.75<1$. On the left, the initial condition is ${m}^{0}={m}^{0,1}=1$ while on the right, the initial condition is a function of space, ${m}^{0}={m}^{0,3}$, and the parameters are defined in TestD.

**Table 1.**In this table, we define all the tests that we show in Section 4.1. The first three rows show the parameters for the accuracy tests for m and $\mathbb{C}$, and the results are summarized in Table 2 and Table 3. The second three rows define the parameters that we use in Figure 2, where we compare the results of changing the diffusivity. The third three rows are the tests showed in Figure 3, varying $gamma$, and the results of the last three rows are in Figure 4, where we change the stabilization parameter $\epsilon $. For the accuracy tests, the number of points of the discretization is specified in Table 2 and Table 3, while for all the other tests, the number of points is fixed, and it is $N=600$.

$\mathit{\alpha}$ | c | D | $\mathit{\epsilon}$ | $\mathit{\gamma}$ | r | ${\mathit{t}}_{\mathbf{fin}}$ | ||
---|---|---|---|---|---|---|---|---|

Accuracy m | TestA: | 0.5 | 1 | 0.01 | - | 0.75 | 0.1 | 1 |

Accuracy $\mathbb{C}$ | TestB: | 1 | 1 | 0.01 | 0.1 | 1.75 | 0.1 | 1 |

Accuracy m | TestC: | 0.5 | 5 | 0.01 | - | 0.75 | 0.01 | 1 |

$D=0.05$ | TestG: | 0.75 | 5 | 0.05 | ${10}^{-3}$ | 0.75 | 0.005 | 15 |

$D=0.01$ | TestD: | 0.75 | 5 | 0.01 | ${10}^{-3}$ | 0.75 | 0.005 | 15 |

$D=0.001$ | TestE: | 0.75 | 5 | 0.001 | ${10}^{-3}$ | 0.75 | 0.005 | 15 |

$\gamma =1$ | TestH: | 0.75 | 5 | 0.01 | ${10}^{-3}$ | 1 | 0.005 | 15 |

$\gamma =0.75$ | TestD: | 0.75 | 5 | 0.01 | ${10}^{-3}$ | 0.75 | 0.005 | 15 |

$\gamma =0.5$ | TestF: | 0.75 | 5 | 0.01 | ${10}^{-3}$ | 0.5 | 0.005 | 15 |

$\epsilon ={10}^{-2}$ | TestI: | 0.75 | 5 | 0.01 | ${10}^{-2}$ | 0.75 | 0.005 | 15 |

$\epsilon ={10}^{-3}$ | TestD: | 0.75 | 5 | 0.01 | ${10}^{-3}$ | 0.75 | 0.005 | 15. |

$\epsilon ={10}^{-4}$ | TestL: | 0.75 | 5 | 0.01 | ${10}^{-4}$ | 0.75 | 0.005 | 15 |

N | Error | Order | N | Error | Order |
---|---|---|---|---|---|

20 | - | - | 20 | - | - |

40 | 0.036030 | - | 40 | 0.036012 | - |

80 | 0.0492860 | −0.4520 | 80 | 0.0493010 | −0.4531 |

160 | 0.01454106 | 1.7610 | 160 | 0.01456192 | 1.7594 |

320 | 0.00690830 | 1.0737 | 320 | 0.00691103 | 1.0752 |

640 | 0.001529779 | 2.1750 | 640 | 0.001528055 | 2.1772 |

N | Error${}_{2}$ | Order |
---|---|---|

25 | - | - |

50 | $9.066\times {10}^{-2}$ | - |

100 | $4.625\times {10}^{-2}$ | 0.97 |

200 | $1.571\times {10}^{-2}$ | 1.56 |

400 | $4.149\times {10}^{-3}$ | 1.92 |

800 | $7.347\times {10}^{-4}$ | 2.50 |

**Table 4.**In this table, we define the two sets of parameters in Equation (40).

${\mathit{\alpha}}_{1},{\mathit{\alpha}}_{2}$ | ${\mathit{c}}_{1},{\mathit{c}}_{2}$ | ${\mathit{D}}_{1}={\mathit{D}}_{2}$ | $\mathit{\epsilon}$ | ${\mathit{\gamma}}_{1},{\mathit{\gamma}}_{2}$ | r | N | ||
---|---|---|---|---|---|---|---|---|

set of parameters | TestM: | 1, 0.5 | $\sqrt{2}$, 1 | 0.1 | ${10}^{-1}$ | 1.75, 0.75 | 0.1 | 600 |

**Table 5.**Here, we show the values of $\mathbb{B}$, after ${n}_{t}$ time steps, where ${n}_{t}={2}^{k},k=0,1,2,3,4,5,6$.

Time | 0.01 | 0.02 | 0.04 | 0.08 | 0.16 | 0.32 | 0.64 |
---|---|---|---|---|---|---|---|

$\left|\right|\mathbb{B}\left|\right|$ | 0.0348 | 0.0538 | 0.0697 | 0.0982 | 0.1509 | 0.2611 | 0.5320 |

$\mathit{\alpha}$ | c | D | $\mathit{\epsilon}$ | $\mathit{\gamma}$ | r | ${\mathit{t}}_{\mathbf{fin}}$ | N | ||
---|---|---|---|---|---|---|---|---|---|

$D=0$ | TestN: | 0.75 | 5 | 0 | ${10}^{-3}$ | 0.75 | 0.005 | 15 | 600 |

$D={10}^{-5}$ | TestO: | 0.75 | 5 | ${10}^{-5}$ | ${10}^{-3}$ | 0.75 | 0.005 | 15 | 600 |

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**MDPI and ACS Style**

Astuto, C.; Boffi, D.; Haskovec, J.; Markowich, P.; Russo, G.
Comparison of Two Aspects of a PDE Model for Biological Network Formation. *Math. Comput. Appl.* **2022**, *27*, 87.
https://doi.org/10.3390/mca27050087

**AMA Style**

Astuto C, Boffi D, Haskovec J, Markowich P, Russo G.
Comparison of Two Aspects of a PDE Model for Biological Network Formation. *Mathematical and Computational Applications*. 2022; 27(5):87.
https://doi.org/10.3390/mca27050087

**Chicago/Turabian Style**

Astuto, Clarissa, Daniele Boffi, Jan Haskovec, Peter Markowich, and Giovanni Russo.
2022. "Comparison of Two Aspects of a PDE Model for Biological Network Formation" *Mathematical and Computational Applications* 27, no. 5: 87.
https://doi.org/10.3390/mca27050087