## 1 Introduction

## 2 Mesh construction

Step | Table | Prerequisite | Method of construction |
---|---|---|---|

1 | VV | parent division | Based on numbering scheme |

2 | EV | VV | Insert edge per VV entry with no duplicates |

3 | FV | parent division | Based on numbering scheme |

4 | VE | EV | Inverse of EV |

5 | VF | FV | Inverse of FV |

6 | EF | EV, VF | Match two faces sharing this edge’s vertices |

7 | FE | EF | Inverse of EF |

8 | FF | EF, FE | Find the other face sharing each edge |

Div | Vertices | Edges | Faces | Avg. edge | Avg. angle | Average area | Edge ratio | Angle ratio | Area ratio |
---|---|---|---|---|---|---|---|---|---|

0 | 12 | 30 | 20 | 63.4 ^{∘} | 72.0 ^{∘} | 6.28 × 10 ^{−1} | 1.00 | 1.00 | 1.00 |

1 | 42 | 120 | 80 | 33.9 ^{∘} | 63.0 ^{∘} | 1.57 × 10 ^{−1} | 1.14 | 1.24 | 1.20 |

2 | 162 | 480 | 320 | 17.2 ^{∘} | 60.8 ^{∘} | 3.93 × 10 ^{−2} | 1.18 | 1.31 | 1.28 |

3 | 642 | 1920 | 1280 | 8.64 ^{∘} | 60.2 ^{∘} | 9.82 × 10 ^{−3} | 1.19 | 1.33 | 1.29 |

4 | 2562 | 7680 | 5120 | 4.33 ^{∘} | 60.0 ^{∘} | 2.45 × 10 ^{−3} | 1.19 | 1.33 | 1.30 |

5 | 10,242 | 30,720 | 20,480 | 2.16 ^{∘} | 60.0 ^{∘} | 6.14 × 10 ^{−4} | 1.19 | 1.33 | 1.30 |

6 | 40,962 | 122,880 | 81,920 | 1.08 ^{∘} | 60.0 ^{∘} | 1.53 × 10 ^{−4} | 1.19 | 1.33 | 1.30 |

7 | 163,842 | 491,520 | 327,680 | 0.54 ^{∘} | 60.0 ^{∘} | 3.84 × 10 ^{−5} | 1.19 | 1.33 | 1.30 |

8 | 655,362 | 1,966,080 | 1,310,720 | 0.27 ^{∘} | 60.0 ^{∘} | 1.16 × 10 ^{−5} | 1.19 | 1.33 | 1.30 |

## 3 Grid blocks

## 4 Representing spherical geometry

t-edge/r-face | \(\varLambda _{1}\) | \(\varLambda _{2}\) | \(\varLambda _{3}\) |
---|---|---|---|

1 | 0 | 1 − δ | δ |

2 | δ | 0 | 1 − δ |

3 | 1 − δ | δ | 0 |

^{∘}. Figure 7 shows the error distribution for element orders one, two, and three. Obviously, the first order element with its planar faces is unable to reproduce the spherical shape resulting in a large error near the center. Switching to the second order element improves the accuracy by three orders of magnitude, while going to third order yields another factor of ∼20. It is evident that both second or third order elements reproduce spherical geometry with remarkable accuracy.

## 5 Evaluation of integrals on a geodesic mesh

### 5.1 Integration on r-edges

### 5.2 Integration on t-edges

### 5.3 Integration on r-faces

### 5.4 Integration on t-faces

### 5.5 Integration on frustums

## 6 Conservative reconstruction on a geodesic mesh

### 6.1 Stencil construction

### 6.2 Utilizing radial similarity

### 6.3 Limiting the reconstruction

## 7 Constrained reconstruction of the magnetic field

### 7.1 Constraint 1

### 7.2 Constraint 2

### 7.3 Constraint 3

### 7.4 Constraint 4

### 7.5 Constraint 5

Order | D(M) | D̄(M) | Unknowns | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|---|---|---|

2 | 4 | 7 | 21 | 3 | 5 | 15 | 12 | 9 |

3 | 10 | 16 | 48 | 9 | 5 | 24 | 30 | 18 |

4 | 20 | 30 | 90 | 19 | 5 | 30 | 60 | 30 |

## 8 Time advance and boundary exchange

## 9 Numerical tests

ρ | e | \(B_{x}\) | |||||||
---|---|---|---|---|---|---|---|---|---|

Division | 5 | 6 | 7 | 5 | 6 | 7 | 5 | 6 | 7 |

\(L_{1}\) | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 |

\(L_{\infty }\) | 2.0 | 2.0 | 1.8 | 2.0 | 2.0 | 2.0 | 1.8 | 1.8 | 1.8 |

ρ | e | \(B_{x}\) | |||||||
---|---|---|---|---|---|---|---|---|---|

Division | 5 | 6 | 7 | 5 | 6 | 7 | 5 | 6 | 7 |

\(L_{1}\) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |

\(L_{\infty }\) | 3.0 | 2.7 | 2.6 | 3.0 | 2.7 | 2.5 | 2.3 | 2.3 | 2.3 |

ρ | e | \(B_{x}\) | |||||||
---|---|---|---|---|---|---|---|---|---|

Division | 5 | 6 | 7 | 5 | 6 | 7 | 5 | 6 | 7 |

\(L_{1}\) | 4.3 | 4.1 | 4.0 | 4.3 | 4.1 | 4.0 | 4.1 | 4.0 | 4.0 |

\(L_{\infty }\) | 4.0 | 3.1 | 3.0 | 4.1 | 3.1 | 2.9 | 3.1 | 3.3 | 3.6 |