# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Derivation of New Compact Schemes

_{1}L

_{1}L

_{2}-Dn

_{2}, where ${n}_{1}$ is an integer equal to the formal order of accuracy of the scheme, ${L}_{1}$ is either C or S for collocated and staggered schemes respectively, ${L}_{2}$ is E, C, or H for explicit, compact, and proposed Hermitian schemes, and ${n}_{2}$ is the order of derivation of the left-hand side of the scheme (0 for interpolation schemes). For instance, Equations (1) and (2) are referred to by the acronyms 4CC-D1 and 4SC-D1 respectively. Moreover, it will be useful to refer to a graphical representation of the various numerical schemes that will be introduced; each variable appearing in the finite-difference formula is depicted with a different symbol, depending on its order of derivation, and in a different position with respect to the stencil, depending on its role in the scheme. The explicit data ${f}_{j}$ are depicted as small circles, while first and successive derivatives are denoted by a corresponding number of inclined dashes. If a variable is a known quantity in the formula, its position in the graphical representation falls within the lower part, below the horizontal line representing the mesh. If it is an unknown, it is positioned above the mesh, in the upper part of the sketch.

#### 2.1. Schemes for Staggered First Derivative

#### 2.2. Schemes for Interpolation

#### 2.3. Schemes for Second Derivative

#### 2.4. Higher-Order Schemes

^{8}f

^{IX}respectively.

## 3. Analysis of Novel Schemes

#### 3.1. Structure of the Schemes

#### 3.2. Resolution Properties

#### 3.3. Evaluation of the Computational Effort

## 4. Application to Incompressible Navier-Stokes Equations

#### 4.1. Spatial Discretization

- Divergence form
- 1
- components i and j of velocity are iterpolated in the direction j and i respectively;
- 2
- the product is performed;
- 3
- the staggered derivative of the product in direction j is computed.

- Advective form
- 1
- components i and j are stagger-differentiated and iterpolated in the directions j and i respectively;
- 2
- the product is performed;
- 3
- the product is interpolated along direction j.

#### 4.2. Time Integration

## 5. Results

#### 5.1. Burggraf Flow

#### 5.2. Periodic Double Mixing Layer

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Modified Wavenumber

## References

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**Figure 4.**Modified wavenumbers/transfer functions relative to several compact and Hermitian schemes, for interpolation, first derivative, and second derivative formulæ. Refer to Table 1 for schemes legend. Solid and dashed lines are relative to novel and classical schemes respectively.

**Figure 6.**Error on ${u}_{1}$ and ${u}_{2}$ as function of the number of meshpoints per direction. The values are reported in Table 3 and are relative to the Burggraf flow testcase.

**Figure 7.**Contour plots of horizontal and vertical velocity component as well as stream function for the Burggraf flow testcase.

**Figure 8.**Iso-contours of vorticity for levels $-6,-4,\dots ,+6$ at time $t=8$ for several schemes and formulations. In all cases, the skew-symmetric form of the convective term is employed and the classical RK4 scheme is used for time integration, unless otherwise specified. The first row shows results for explicit schemes of increasing accuracy on a collocated layout; the second row refers to classical compact schemes on a staggered layout; the third row to the novel Hermitian schemes on a staggered layout.

**Figure 9.**Time evolution of kinetic energy for all the methods presented in Figure 8, which also serves as a reference for the labels.

Curve Label | Figure 4a | Figure 4b | Figure 4c |
---|---|---|---|

a | 4CE-D1 | 4SE-D0 | 4CE-D2 |

b | 4CC-D1 (1) | 4CC-D2 (12) | |

c | 4SC-D1 (2) | 4SC-D0 (10) | |

d | 6SC-D1 (17) | 6SC-D0 (20) | 5CC-D2 (24) |

e | 8SC-D1 (18) | 8SC-D0 (21) | 7CC-D2 (25) |

f | 4SH-D1 (7) | 4SH-D0 (9) | 4CH-D2 (11) |

g | 6SH-D1 (15) | 6SH-D0 (8) | 6CH-D2 (22) |

h | 8SH-D1 (16) | 8SH-D0 (19) | 8CH-D2 (23) |

**Table 2.**Resolving efficiency e and integral resolving efficiency ${e}_{\mathrm{I}}^{}$ for selected classical and new first derivation schemes.

Scheme | e—Equation (33) | ${\mathit{e}}_{\mathbf{I}}^{}$—Equation (34) | |||
---|---|---|---|---|---|

$\mathit{\epsilon}=0.1$ | $\mathit{\epsilon}=0.01$ | $\mathit{\epsilon}=0.001$ | $\mathit{k}=1$ | $\mathit{k}=2$ | |

4CE-D1 | 0.444 | 0.240 | 0.133 | 0.540 | 0.135 |

4CC-D1 (Equation (1)) | 0.594 | 0.355 | 0.205 | 0.668 | 0.402 |

4SC-D1 (Equation (2)) | 0.782 | 0.432 | 0.243 | 0.915 | 0.965 |

6SC-D1 (Equation (17)) | 0.902 | 0.612 | 0.421 | 0.954 | 0.986 |

8SC-D1 (Equation (18)) | 0.950 | 0.702 | 0.530 | 0.969 | 0.993 |

4SH-D1 (Equations (1) and (7)) | 1.000 | 0.468 | 0.260 | 0.977 | 0.998 |

6SH-D1 (Equations (13) and (15)) | 1.000 | 0.601 | 0.405 | 0.981 | 0.998 |

8SH-D1 (Equations(14) and (16)) | 1.000 | 0.678 | 0.499 | 0.985 | 0.999 |

**Table 3.**Grid refinement study for Burggraf flow for ${u}_{1}$ and ${u}_{2}$. The values are depicted in Figure 6.

${\mathit{u}}_{1}$ | ${\mathit{u}}_{2}$ | |||
---|---|---|---|---|

N | Error | Order | Error | Order |

16 | 4.167 × 10^{−4} | — | 4.683 × 10^{−4} | — |

32 | 1.812 × 10^{−5} | 4.523 | 2.730 × 10^{−5} | 4.100 |

64 | 6.333 × 10^{−7} | 4.838 | 1.066 × 10^{−6} | 4.677 |

128 | 2.564 × 10^{−8} | 4.626 | 3.752 × 10^{−8} | 4.829 |

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**MDPI and ACS Style**

De Angelis, E.M.; Coppola, G.; Capuano, F.; De Luca, L.
Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations. *Appl. Sci.* **2018**, *8*, 1066.
https://doi.org/10.3390/app8071066

**AMA Style**

De Angelis EM, Coppola G, Capuano F, De Luca L.
Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations. *Applied Sciences*. 2018; 8(7):1066.
https://doi.org/10.3390/app8071066

**Chicago/Turabian Style**

De Angelis, Enrico Maria, Gennaro Coppola, Francesco Capuano, and Luigi De Luca.
2018. "Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations" *Applied Sciences* 8, no. 7: 1066.
https://doi.org/10.3390/app8071066