# Implementation of an Implicit Solver in ADCIRC Storm Surge Model

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Methodology

## 4. Results and Discussion

#### 4.1. Solvers Stability

#### 4.2. Water Elevation and Velocity Comparison

- For each timestep, the differences of elevation and velocity are calculated for all wet nodes of the mesh by subtracting elevation and velocity of the second solver, say Case 5, from those of the first solver, say Case 1 (e.g., h_diff = h_Case1—h_Case5; V_diff = V_Case1—V_Case5).
- The average and standard deviation for the above differences of water elevation and velocity are calculated for each timestep.
- Maximums and minimums of the above differences between the results of the two solvers are obtained to identify the worst node-to-node differences for each timestep.

#### 4.3. Impact of Timestep

#### 4.4. Buoys Time Series Comparison

#### 4.5. High Water Mark Comparison

^{2}) value of 0.666 is obtained, similar to the ones reported in [5]. This value of R

^{2}is considered very good in perspective since even a sophisticated mesh and model setup for Hurricane Ike, which had the maximum water elevation of 5 m, the best fit for ADCIRC produced R

^{2}value of 0.716 [26]. Most importantly, the implicit solver (Case 5) gives almost identical results to the ones produced by the lumped explicit (Case 1) and semi-implicit (Case 3) solvers, as shown in Figure 9b,c.

#### 4.6. Execution Time

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Maximum elevation and velocity comparison study: (

**a**) Elevation of Lumped Explicit (Case 1) vs. Implicit (Case 5); (

**b**) velocity magnitude of Lumped Explicit (Case 1) vs. Implicit (Case 5); (

**c**) elevation of Semi-Implicit (Case 3) vs. Implicit (Case 5); (

**d**) velocity magnitude of Semi-Implicit (Case 3) vs. Implicit (Case 5).

**Figure 2.**Time snapshot differences of simulated water elevation and velocity magnitude at 10 a.m. on 29 August 2005 UTC between: (

**a**) Lumped Explicit (Case 1) vs. Implicit (Case 5); (

**b**) Semi-Implicit (Case 3) vs. Implicit (Case 5).

**Figure 3.**Time series average (‘Ave’), standard deviation (‘StDev’), minimum (‘Min’) and maximum (‘Max’) of water elevation and velocity differences between: (

**a**–

**d**) Lumped Explicit (Case 1) vs. Implicit (Case 5), and (

**e**–

**h**) Semi-Implicit (Case 3) vs. Implicit (Case 5).

**Figure 4.**Maximum elevation and velocity comparison for timestep size study: (

**a**) Elevation of Implicit (Case 5) vs. Implicit (Case 6); (

**b**) velocity magnitude of Implicit (Case 5) vs. Implicit (Case 6).

**Figure 5.**Implicit solver (Case 5) vs. Implicit solver (Case 6): (

**a**) Time snapshot differences of water elevation and velocity magnitude at 10 a.m. on 29 August 2005 UTC. (

**b**) Maximum elevation and velocity differences.

**Figure 6.**Impact of timesteps on Implicit solver between small (Case 5) and large (Case 6) steps: (

**a**) Elevation time series average and standard deviation; (

**b**) minimum and maximum of water elevation; (

**c**) velocity time series average and standard deviation; (

**d**) minimum and maximum of water velocity.

**Figure 8.**Observed time series data vs. modeled time series results of Hurricane Katrina storm surge: (

**a**) Station ID 8735180 Dauphin Island AL; (

**b**) Station ID 8735180 Pilots Station East SW Pass LA; (

**c**) Station ID 8747766 Waveland MS; and (

**d**) Station ID 8761724 Grand Isle.

**Figure 9.**Hurricane Katrina HWM comparisons: (

**a**) Observed data vs. Implicit Solver (Case 5); (

**b**) Lumped Explicit Solver (Case 1) vs. Implicit Solver (Case 5); (

**c**) Semi-Implicit Solver (Case 3) vs. Implicit Solver (Case 5).

**Figure 10.**Execution times comparison between Lumped Explicit, Semi-Implicit, and Implicit solvers using Hurricane Katrina hindcast: (

**a**) Execution time using the same timestep (Cases 1, 3, and 5) for all solvers (see Table 3); (

**b**) execution time using maximum timesteps (Cases 2, 4, and 6) to each solver (see Table 4).

Timestep (s) | Lumped Explicit | Semi-Implicit | Implicit | ||||
---|---|---|---|---|---|---|---|

Comment | Walltime (s) | Comment | Walltime (s) | Comment | Walltime (s) | Nonlinear Iterations | |

2 | Success | 1283 | Success | 1771 | Success | 10,466 | 2 |

4 | Success | 902 | Success | 1130 | Success | 5712 | 2 |

8 | Fail | N/A | Success | 866 | Success | 3497 | 2 |

12 | Fail | N/A | Success | 2685 | 2 | ||

40 | Success | 1368 | 2 | ||||

100 | Success | 1213 | 3 | ||||

120 | Success | 1272 | 4 | ||||

150 | Fail | N/A | 6 |

Case # | Solver | Timestep (s) |
---|---|---|

1 | Lumped Explicit | 2 |

2 | Lumped Explicit | 4 |

3 | Semi-Implicit | 2 |

4 | Semi-Implicit | 8 |

5 | Implicit | 2 |

6 | Implicit | 120 |

No. of CPUs | ADCIRC Lumped Explicit Case 1 | ADCIRC Semi-Implicit Case 3 | ADCIRC Implicit Case 5 |
---|---|---|---|

1 | 58,591 | 79,686 | 572,863 |

2 | 35,390 | 72,672 | 500,688 |

4 | 20,571 | 39,423 | 276,973 |

8 | 10,662 | 20,487 | 131,750 |

16 | 5837 | 10,149 | 80,415 |

32 | 3224 | 5438 | 42,342 |

64 | 1951 | 2855 | 19,901 |

128 | 1283 | 1771 | 10,466 |

256 | 1097 | 1440 | 7657 |

No. of CPUs | ADCIRC Lumped Explicit Case 2 | ADCIRC Semi-Implicit Case 4 | ADCIRC Implicit Case 6 |
---|---|---|---|

1 | 25,062 | 21,970 | 52,528 |

2 | 21,693 | 20,107 | 41,529 |

4 | 11,531 | 11,083 | 21,747 |

8 | 6024 | 5267 | 9467 |

16 | 2958 | 2539 | 5942 |

32 | 1930 | 1750 | 3364 |

64 | 1259 | 1104 | 2128 |

128 | 902 | 866 | 1272 |

256 | 602 | 785 | 1142 |

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**MDPI and ACS Style**

Alghamdi, A.; Akbar, M.K. Implementation of an Implicit Solver in ADCIRC Storm Surge Model. *J. Mar. Sci. Eng.* **2018**, *6*, 62.
https://doi.org/10.3390/jmse6020062

**AMA Style**

Alghamdi A, Akbar MK. Implementation of an Implicit Solver in ADCIRC Storm Surge Model. *Journal of Marine Science and Engineering*. 2018; 6(2):62.
https://doi.org/10.3390/jmse6020062

**Chicago/Turabian Style**

Alghamdi, Abdullah, and Muhammad K. Akbar. 2018. "Implementation of an Implicit Solver in ADCIRC Storm Surge Model" *Journal of Marine Science and Engineering* 6, no. 2: 62.
https://doi.org/10.3390/jmse6020062