# Implementation of a Local Time Stepping Algorithm and Its Acceleration Effect on Two-Dimensional Hydrodynamic Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Global Time Stepping Scheme

#### 2.1.1. Governing Equation

#### 2.1.2. Numerical Technique

#### 2.2. Local Time Stepping Scheme

#### 2.2.1. Reconstructing the Local Time Step of Each Cell

#### 2.2.2. Calculation of Element Variables at the Interface

## 3. Numerical Tests

#### 3.1. Anti-Symmetric Dam Break Case

^{−3}to 10

^{−2}range. At the beginning of the calculation, the total water volume was 3 × 10

^{5}m

^{3}. When the value of L was 2, 3, and 4, the relative error of the water volume was 0.0017%, 0.0019%, and 0.0045%, respectively, which indicates that the simulation meets the accuracy and conservation of water volume requirements.

#### 3.2. Non-Flat Bottom Dam Break Case

^{3}, and the final water volume was 959.978 m

^{3}. The relative error was 0.01%, which is within the discrete error range. The model dealt well with wetting and drying fronts over the complex terrain, satisfied computational stability, and conservation of water volume requirements, and provided results that were consistent with previous experimental results [38,39].

#### 3.3. Navigable Flow Simulation Case

## 4. Discussion

#### 4.1. The Influence of the Proportion of Refined Mesh on the Acceleration Effect

^{2}). The spatial resolution level varied from 1 to 6 m, with a minimum resolution close to the breach of 1 m, surrounded by a smooth transition up to a maximum resolution of 6 m in the outer domain. The mesh generation results for the refined areas for various proportions in the anti-symmetric dam break case is shown in Figure 13. The numerical simulation was performed for refined areas of 5%, 10%, 25%, 50%, and 75%. The impact of increasing the local refinement area on the acceleration effect was analyzed for a time-step level of m = 0 in the refinement area, m = 2 in the coarse grid area, and m = 1 in the buffer zone.

^{2}to 73,306 at 3 hm

^{2}, which represents an increase by a factor of 5.5. The operating time of the model rose from 608 to 3394 s, which represents an increase by a factor of 5.6. The number of grids is a key factor determining the overhead of computing. There is a direct positive correlation between the number of grids and computing overheads.

#### 4.2. The Impact of the Different Scale of the Mesh on Acceleration Effect

## 5. Conclusions

- (1)
- Based on the FVM for unstructured grids, an LTS algorithm was implemented that improved the computational efficiency of the model, while satisfying water conservation conditions. In the anti-symmetric dam break case, a speedup ratio of 2.1 was achieved, which saved 53% in execution time. The speedup ratio of the non-flat bottom dam break case was 1.3, which represented a shortening of 26% in the calculation time. The numerical simulation of the navigable flow of the river reach between the Three Gorges and Gezhouba Dams achieved a speedup ratio of 1.9, which represented a saving of 49% in modeling time.
- (2)
- The proportions of coarse to refined meshes on the acceleration effect of the LTS algorithm were noticeable. It was evident that a higher speedup ratio was obtained when the proportion of the refined mesh was minimized. When the proportion of the refined mesh was high, the acceleration effect was not significant. It is clear that the LTS algorithm is best suited to situations in which refinement is only required in small regions.
- (3)
- When using the LTS algorithm on non-uniform unstructured grids, the larger the grid scale difference, the more obvious the grid layering became. This led to increased acceleration effects. However, computational accuracy was slightly impaired by excessive differences in grid mesh size.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the two-dimensional model with finite volume discretization: physical variables are defined at the grid center, where ${\mathrm{\Omega}}_{k}$ is the kth control volume; $\mathit{n}$ is the unit normal vector; $p\left(i,j\right)$ is a grid point; and $\Delta {l}_{kj}$ is the length of a side.

**Figure 2.**Flow diagram showing global time stepping (GTS) and local time stepping (LTS) algorithms. In the GTS scheme, each cell is updated with the same time step $\Delta t$, while in the LTS scheme, a locally allowable maximum time step ($\Delta {t}_{i}^{b}$) is used for updating variables in each cell.

**Figure 3.**Example of the m-level assignment to each cell in the unstructured grid. (

**a**) Distribution showing initial values assigned by Equation (10); (

**b**) distribution after being modified by Equation (11a), with the reassignment of buffer zone values based on neighboring cells values.

**Figure 4.**Sketch showing the calculation of element variables at an interface: the variable $\mathrm{U}\left(h,u,v\right)$ refers to the water depth $h$, and the velocity components $u$ and $v$ in the $x$ and $y$ directions, respectively. Within a maximum time step of $\Delta {t}_{max}$ = 0.04 s, the coarse mesh areas (time-step level, m = 2) were updated only once. The value of ${\mathrm{U}}_{1}\left({h}_{1},{u}_{1},{v}_{1}\right)$ at 0 s was updated to ${\mathrm{U}}_{1}^{*}\left({h}_{1}^{*},{u}_{1}^{*},{v}_{1}^{*}\right)$ after 0.04 s. The blank value at 0.02 s was estimated using the average of the values at the former and the latter time-step level, giving ${\mathrm{U}}_{1}^{\prime}\left(\frac{{h}_{1}+{h}_{1}^{*}}{2},\frac{{u}_{1}+{u}_{1}^{*}}{2},\frac{{v}_{1}+{v}_{1}^{*}}{2}\right)$. Immediately following this, the intermediate grids (m = 1) were updated twice over this maximum time step. The values of variables at 0.01 and 0.03 s were similarly approximated. Finally, the refined mesh grids (m = 0) were updated four times over this interval. All variables were ultimately updated to 0.04 s, which allowed the next time step to be performed.

**Figure 5.**Sketch showing the plane geometry of an anti-symmetric dam break, with the breach occurring in the center.

**Figure 6.**Details of the non-uniform unstructured meshes and time-step level (m-level) distribution for the anti-symmetric dam break case.

**Figure 7.**Results of the simulation using the global time stepping strategy at dam break (t = 7.2 s). (

**a**) Three-dimensional visualization of the water surface; (

**b**) contour plot of water depth; (

**c**) velocity vector distribution.

**Figure 8.**Computational cost (s) and speedup ratios for different local time-step rank values (L) for the anti-symmetric dam break case. As the value of L increases, the computational cost decreases and the speedup ratio increases.

**Figure 9.**Details of the non-uniform unstructured meshes, time-step level (m-level) distribution, and bathymetric point arrangement for the non-flat bottom dam break case.

**Figure 10.**Three-dimensional visualization of the inundation process at various time points following the dam break. (

**a**) t = 0 s; (

**b**) t = 10 s; (

**c**) t = 30 s; (

**d**) t = 100 s. (GTS: global time stepping strategy, while L is the time-step rank value in the local time stepping strategy).

**Figure 11.**Comparison of water levels at different local time-step rank values (L) for the non-flat bottom dam break case (GTS: global time stepping strategy).

**Figure 12.**Computational cells and details of refined areas for the river reach between the Three Gorges and Gezhouba Dams.

**Figure 13.**Locally refined areas of various proportions (%) in the anti-symmetric dam break case: the refined area of (

**a**) 5%; (

**b**) 10%; (

**c**) 25%; (

**d**) 50%, (

**e**) 75%.

**Figure 14.**Speedup ratio for implementing the LTS algorithm for various refined area proportions in the dam break model for different local time-step rank values (L). As the refined area increased, the speedup ratio decreased. The acceleration effect increased with increasing L.

**Table 1.**Mean square errors of water depth ($h$) and velocity components ($u$, $v$) for different local time-step rank values (L) in the anti-symmetric dam break case.

t(s) | L = 2 | L = 3 | L = 4 | ||||||
---|---|---|---|---|---|---|---|---|---|

L_{s}(u) × 10^{−2} | L_{s}(v) × 10^{−2} | L_{s}(h) × 10^{−2} | L_{s}(u) × 10^{−2} | L_{s}(v) × 10^{−2} | L_{s}(h) × 10^{−2} | L_{s}(u) × 10^{−2} | L_{s}(v) × 10^{−2} | L_{s}(h) × 10^{−2} | |

7.2 | 0.41 | 0.38 | 0.22 | 0.65 | 0.56 | 0.50 | 1.07 | 0.71 | 0.79 |

15.2 | 0.34 | 0.31 | 0.17 | 0.65 | 0.64 | 0.45 | 1.02 | 0.67 | 0.87 |

23.2 | 0.50 | 0.43 | 0.39 | 1.02 | 0.81 | 1.07 | 1.84 | 1.20 | 1.99 |

31.2 | 0.66 | 0.72 | 0.57 | 1.53 | 1.45 | 1.39 | 2.81 | 2.63 | 2.62 |

39.2 | 0.74 | 0.75 | 0.29 | 1.34 | 1.37 | 0.62 | 2.43 | 2.40 | 1.04 |

47.2 | 0.55 | 0.70 | 0.30 | 0.89 | 1.24 | 0.72 | 2.03 | 3.31 | 1.63 |

55.2 | 0.55 | 0.71 | 0.36 | 0.84 | 1.06 | 0.96 | 1.72 | 2.16 | 1.76 |

63.2 | 0.51 | 0.60 | 0.31 | 0.95 | 1.18 | 0.86 | 1.63 | 3.12 | 2.14 |

71.2 | 0.71 | 0.70 | 0.30 | 1.17 | 1.25 | 0.68 | 2.82 | 2.81 | 1.46 |

79.2 | 0.64 | 0.63 | 0.27 | 1.10 | 1.05 | 0.74 | 2.54 | 2.32 | 1.50 |

120 | 0.67 | 0.63 | 0.18 | 1.33 | 1.12 | 0.47 | 2.69 | 2.26 | 1.10 |

160 | 0.61 | 0.56 | 0.24 | 1.03 | 0.97 | 0.70 | 2.35 | 2.26 | 1.85 |

average | 0.57 | 0.59 | 0.30 | 1.04 | 1.06 | 0.76 | 2.08 | 2.15 | 1.56 |

**Table 2.**Computing time T (s), time-saving ratio ${T}_{r}$ (%), speedup ratio (${S}_{n}$) of different local time-step rank values (L) for the anti-symmetric dam break case (GTS: global time stepping strategy).

Test | T | ${\mathit{T}}_{\mathit{r}}$ | ${\mathit{S}}_{\mathit{n}}$ |
---|---|---|---|

GTS | 608 | - | - |

L = 2 | 349 | 43 | 1.74 |

L = 3 | 295 | 51 | 2.06 |

L = 4 | 285 | 53 | 2.13 |

**Table 3.**Computing time T (s), time saving ratio ${T}_{r}$ (%), speedup ratio (${S}_{n}$) for different local time-step rank values (L) for the non-flat bottom dam break case (GTS: global time stepping strategy).

Test | T | ${\mathit{T}}_{\mathit{r}}$ | ${\mathit{S}}_{\mathit{n}}$ |
---|---|---|---|

GTS | 148 | - | - |

L = 2 | 115 | 22 | 1.28 |

L = 3 | 110 | 26 | 1.34 |

**Table 4.**Simulation results for the navigable flow of the waterway between the Three Gorges and Gezhouba Dams.

Data | River Reach | Water Level (m) | Flow Velocity (m/s) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Observation Time | Discharge (m ^{3}/s) | Measured | Calculated | Measured | Calculated | ||||||||||

GTS | L = 2 | L = 3 | L = 4 | L = 5 | GTS | L = 2 | L = 3 | L = 4 | L = 5 | ||||||

2008.08.20 | 28,400 | Letianxi | Value | 68.32 | 68.39 | 68.39 | 68.39 | 68.39 | 68.39 | 1.56 | 1.63 | 1.63 | 1.63 | 1.63 | 1.63 |

Deviation | - | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | - | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | |||

2008.08.20 | 29,700 | Liantuo | Value | 68.35 | 68.41 | 68.41 | 68.41 | 68.41 | 68.41 | 2.13 | 2.16 | 2.16 | 2.16 | 2.16 | 2.16 |

Deviation | - | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 | - | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | |||

2008.09.04 | 31,700 | Shipai | Value | 67.83 | 67.90 | 67.90 | 67.90 | 67.90 | 67.90 | 1.61 | 1.55 | 1.55 | 1.55 | 1.55 | 1.55 |

Deviation | - | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | - | −0.06 | −0.06 | −0.06 | −0.06 | −0.06 | |||

T | 6.65 | 3.86 | 3.56 | 3.45 | 3.39 | ||||||||||

T_{r} | - | 41.96 | 46.47 | 48.06 | 48.96 | ||||||||||

S_{n} | - | 1.72 | 1.87 | 1.93 | 1.96 |

Refined Area Proportion | Refined Area (hm^{2}) | Total Grid Number | Test | |||
---|---|---|---|---|---|---|

Index | GTS | L = 2 | L = 3 | |||

5% | 0.2 | 13,338 | T | 608 | 467 | 436 |

${T}_{r}$ | - | 23% | 28% | |||

${S}_{n}$ | - | 1.30 | 1.39 | |||

10% | 0.4 | 18,986 | T | 857 | 727 | 702 |

${T}_{r}$ | - | 15% | 18% | |||

${S}_{n}$ | - | 1.18 | 1.22 | |||

25% | 1 | 32,230 | T | 1474 | 1328 | 1308 |

${T}_{r}$ | - | 9.8% | 11.3% | |||

${S}_{n}$ | - | 1.11 | 1.13 | |||

50% | 2 | 48,622 | T | 2234 | 2197 | 2183 |

${T}_{r}$ | - | 1.6% | 2.3% | |||

${S}_{n}$ | - | 1.01 | 1.02 | |||

75% | 3 | 73,306 | T | 3393 | 3376 | 3352 |

${T}_{r}$ | - | 0.5% | 0.5% | |||

${S}_{n}$ | - | 1.01 | 1.01 |

Spatial Resolution | Grid Number | Test | ||
---|---|---|---|---|

Index | GTS | LTS | ||

1–2 m | 35,104 | T | 1558 | 1223 |

${T}_{r}$ | - | 22% | ||

${S}_{n}$ | - | 1.27 | ||

1–3 m | 22,084 | T | 973 | 769 |

${T}_{r}$ | - | 21% | ||

${S}_{n}$ | - | 1.26 | ||

1–4 m | 17,108 | T | 757 | 589 |

${T}_{r}$ | - | 22% | ||

${S}_{n}$ | - | 1.28 | ||

1–5 m | 14,380 | T | 641 | 494 |

${T}_{r}$ | - | 23% | ||

${S}_{n}$ | - | 1.30 | ||

1–6 m | 13,338 | T | 608 | 436 |

${T}_{r}$ | - | 28% | ||

${S}_{n}$ | - | 1.39 |

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**MDPI and ACS Style**

Yang, X.; An, W.; Li, W.; Zhang, S. Implementation of a Local Time Stepping Algorithm and Its Acceleration Effect on Two-Dimensional Hydrodynamic Models. *Water* **2020**, *12*, 1148.
https://doi.org/10.3390/w12041148

**AMA Style**

Yang X, An W, Li W, Zhang S. Implementation of a Local Time Stepping Algorithm and Its Acceleration Effect on Two-Dimensional Hydrodynamic Models. *Water*. 2020; 12(4):1148.
https://doi.org/10.3390/w12041148

**Chicago/Turabian Style**

Yang, Xiyan, Wenjie An, Wenda Li, and Shanghong Zhang. 2020. "Implementation of a Local Time Stepping Algorithm and Its Acceleration Effect on Two-Dimensional Hydrodynamic Models" *Water* 12, no. 4: 1148.
https://doi.org/10.3390/w12041148