# A Three-Order, Divergence-Free Scheme for the Simulation of Solar Wind

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Mesh Grid System and Numerical Scheme Formulation

#### 3.1. Reconstruction for Cell Average Variables over the Cell

#### 3.2. Reconstruction for Face Average Magnetic Fields at Cell Faces

#### 3.3. The Third Order Reconstruction for Magnetic Fields over a Cell

- (1)
- ${B}_{r}(r,\theta ,\varphi )$ exactly matches ${b}_{r,im}(\theta ,\varphi )$ and ${b}_{r,ip}(\theta ,\varphi )$ on $(im,j,k)$ and $(ip,j,k)$ faces;
- (2)
- ${B}_{\theta}(r,\theta ,\varphi )$ exactly matches ${b}_{\theta ,jm}(r,\varphi )$ and ${b}_{\theta ,jp}(r,\varphi )$ on $(i,jm,k)$ and $(i,jp,k)$ faces;
- (3)
- ${B}_{\varphi}(r,\theta ,\varphi )$ exactly matches ${b}_{\varphi ,km}(r,\theta )$ and ${b}_{\varphi ,kp}(r,\theta )$ on $(i,j,km)$ and $(i,j,kp)$ faces;
- (4)
- Any coefficients in ${B}_{r}(r,\theta ,\varphi ),{B}_{\theta}(r,\theta ,\varphi )$ and ${B}_{\varphi}(r,\theta ,\varphi )$ that remain as free parameters are set to zero.

#### 3.4. Three-Order Runge–Kutta Scheme

#### 3.5. Initial-Boundary Value Conditions

- Solving Equation (11) using the least-squares method based on the singular value decomposition (SVD);
- The magnetic field at face centers are set as arithmetic average of the two states at the interface, for example, ${\widehat{\mathbf{B}}}_{ip,j,k}=\left({\widehat{\mathbf{B}}}_{i+1,j,k}+{\widehat{\mathbf{B}}}_{i,j,k}\right)\times 0.5$, ${\widehat{\mathbf{B}}}_{i+1,j,k}$ and ${\widehat{\mathbf{B}}}_{i,j,k}$ is the face average of the reconstructed polynomial ${\mathbf{B}}_{i+1,j,k}$ and ${\mathbf{B}}_{i,j,k}$ at face $(ip,j,k)$.
- If the $\begin{array}{cc}\hfill {\overline{\nabla \xb7\mathbf{B}}}_{i,j,k}=& \frac{\Delta {S}_{r,ip,j,k}{\widehat{\mathbf{B}}}_{ip,j,k}-\Delta {S}_{r,im,j,k}{\widehat{\mathbf{B}}}_{im,j,k}+\Delta {S}_{\theta ,i,jp,k}{\widehat{\mathbf{B}}}_{i,jp,k}-\Delta {S}_{\theta ,i,jm,k}{\widehat{\mathbf{B}}}_{i,jm,k}}{\Delta {V}_{i,j,k}}\hfill \\ & +\frac{\Delta {S}_{\varphi ,i,j,kp}{\widehat{\mathbf{B}}}_{i,j,kp}-\Delta {S}_{\varphi ,i,j,kp}{\widehat{\mathbf{B}}}_{i,j,km}}{\Delta {V}_{i,j,k}}.\le {10}^{-10}\hfill \end{array}$, end. Else, go to 1.

## 4. Numerical Results

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The model results for the magnetic field lines, radial speed ${v}_{r}$ (km/s), and number density $N(lo{g}_{10}/{\mathrm{cm}}^{3})$ on the meridional plane of $\varphi ={180}^{\circ}\u2013{0}^{\circ}$ (

**top**) and $\varphi ={270}^{\circ}\u2013{90}^{\circ}$ (

**bottom**) from 1 to 20 ${\mathsf{R}}_{s}$.

**Figure 3.**Coronal polarized brightness images from 2.0 R${}_{s}$ to 6 R${}_{s}$, and observed by LASCO C2/SOHO (

**bottom**). The (

**left**,

**right**) panels of the top row are the views projected on the meridional planes with $\varphi ={180}^{\circ}\u2013{0}^{\circ}$ and $\varphi ={270}^{\circ}\u2013{90}^{\circ}$, respectively. The (

**bottom**) panels are the observations made on 14 August and 5 August 2018.

**Figure 4.**The model results for radial speed ${v}_{r}$ (km/s) and Alfvénic surface (white line) on the meridional plane of $\varphi ={180}^{\circ}\u2013{0}^{\circ}$ (

**left**) and $\varphi ={270}^{\circ}\u2013{90}^{\circ}$ (

**right**) from 1–20 R${}_{s}$.

**Figure 5.**The initial profile of $\mathrm{Error}1\left(\mathbf{B}\right)$ (

**left**) and $\mathrm{Error}2\left(\mathbf{B}\right)$ (

**right**) on the meridional plane of $\varphi ={180}^{\circ}\u2013{0}^{\circ}$ from 1 to 20 R${}_{s}$.

**Figure 6.**The profile of $\mathrm{Error}1\left(\mathbf{B}\right)$ (

**left**) and $\mathrm{Error}2\left(\mathbf{B}\right)$ (

**right**) on the meridional plane of $\varphi ={180}^{\circ}\u2013{0}^{\circ}$ from 1 to 20 R${}_{s}$ at t = 20 h.

**Figure 7.**The L1 normalization for divergence of magnetic fields on the meridional plane of $\varphi ={180}^{\circ}\u2013{0}^{\circ}$ from 1 to 20 R${}_{s}$ at t = 20 h.

**Figure 8.**The temporal evolution of the $lo{g}_{10}\mathrm{Error}1{\left(\mathbf{B}\right)}^{ave}$ and $lo{g}_{10}\mathrm{Error}2{\left(\mathbf{B}\right)}^{ave}$ in the calculation.

**Figure 9.**The number density $N({log}_{10}/{\mathrm{cm}}^{3})$ distribution (

**left**) and radial speed ${v}_{r}$ (km/s) profiles (

**right**) along heliocentric distance with different latitudes $\theta ={2}^{\circ}$ and $\theta ={90}^{\circ}$.

**Figure 10.**The modeled results at 2.5 R${}_{s}$ (

**top row**) and 20 R${}_{s}$ (

**bottom row**). The left column denotes number density N with unit ${10}^{6}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cm}}^{-3},{10}^{4}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cm}}^{-3}$ from top to bottom, the middle column denotes radial speed ${v}_{r}$ with unit km/s, the last column denotes temperature T with unit ${10}^{5}$ K.

**Figure 11.**Temporal profiles of radial speed ${v}_{r}$ with unit km/s from the mapped observations (solid line) and the MHD model (dashed line).

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**MDPI and ACS Style**

Zhang, M.; Feng, X.
A Three-Order, Divergence-Free Scheme for the Simulation of Solar Wind. *Universe* **2022**, *8*, 371.
https://doi.org/10.3390/universe8070371

**AMA Style**

Zhang M, Feng X.
A Three-Order, Divergence-Free Scheme for the Simulation of Solar Wind. *Universe*. 2022; 8(7):371.
https://doi.org/10.3390/universe8070371

**Chicago/Turabian Style**

Zhang, Man, and Xueshang Feng.
2022. "A Three-Order, Divergence-Free Scheme for the Simulation of Solar Wind" *Universe* 8, no. 7: 371.
https://doi.org/10.3390/universe8070371