# A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Method

#### 3.1. FVM on Triangular Grids

#### 3.2. Approximated Riemann Solver

#### 3.3. Treatment of Source Term

#### 3.4. Time Integration Scheme

#### Stability Criteria

#### 3.5. Second-Order Spatial Reconstructing and Limiting Method

#### 3.5.1. A Review of Monoslope MUSCL Scheme

**U**and ${T}_{i}{M}_{ij}$ denotes the direction vector from ${T}_{i}$ to ${M}_{ij}$. The value of ${U}_{L}^{M}$ can be extrapolated from the centroid ${T}_{i}$ with a limited slope:

#### 3.5.2. Review of Multislope MUSCL Scheme

#### 3.5.3. A Novel Multislope MUSCL Scheme

#### 3.6. Treatment of the Wet-Dry Interface

_{L}, so spurious flow motion is effectively prevented and the well-balanced property is perfectly preserved.

#### 3.7. Boundary Conditions

#### 3.7.1. Liquid Boundary

#### 3.7.2. Solid Boundary

## 4. Verification and Application

#### 4.1. Stationary Flow with Wet-Dry Interface

#### 4.2. Potential Flow over Uneven Bottom

#### 4.3. Water Sloshing in a Parabolic Basin

#### 4.4. Steady Flow over Frictional Uneven Bottom

#### 4.5. Flow in a Compound Channel

#### 4.6. Flow in a Contracting-Expanding Channel over Uneven Bottom

^{3}/s at the inlet boundary, and the outlet boundary conditions do not influence the flow regime upstream. The water surface along the path is presented in Figure 21a. Zone 1 denotes the converging section where the cross-section contracts over a flat bottom to create a critical depth; zone 2 denotes the throat section over a sloping bottom where supercritical flow occurs; and zone 3 denotes the diverging section over an adverse slope bottom where the flow is supercritical. The flow regime along the flume varies from subcritical to critical and eventually reaches supercritical conditions downstream of the throat section. Complexly varying flow regimes present a great challenge to the accuracy and stability of the simulation. A robust numerical scheme should provide an accurate prediction for different flow regimes. The parameters of this experiment are listed in Table 2.

#### 4.7. A Symmetric 2D Riemann Problem

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 6.**Comparisons between numerical and analytical results at t = 1500 s: (

**a**) water surface; (

**b**) unit width discharges in x and y directions.

**Figure 9.**Numerical results of the potential flow test: (

**a**) contours of water depth; (

**b**) vector field of flow velocities.

**Figure 11.**Comparison between simulated and analytical water surface for the mesh of ∆L = 55.18 m. (

**a**) t = T; (

**b**) t = T + T/3; (

**c**) t = T + T/2; (

**d**) t = 2T + T/6.

**Figure 12.**Comparison between simulated and analytical velocities for the mesh of ∆L = 55.18 m at the point (−5000, −100): (

**a**) u velocity; (

**b**) v velocity.

**Figure 14.**Errors of the numerical scheme in ${L}_{1}$ and ${L}_{2}$ norms at t = 3T for $h$, ${q}_{x}$ and ${q}_{y}$: (

**a**) ${L}_{1}$ errors; (

**b**) ${L}_{2}$ errors.

**Figure 15.**Four triangular mesh types used for computation: (

**a**) Delaunay; (

**b**) orthogonal-I; (

**c**) orthogonal-II; (

**d**) distorted.

**Figure 17.**Comparisons between simulated and analytical water surface on the distorted mesh along: (

**a**) the diagonal section; (

**b**) the horizontal-symmetry section.

**Figure 21.**Comparison of experimental and simulated results along the flume: (

**a**) water surface; (

**b**) x-directional velocity.

**Figure 24.**Numerical results of the 2D Riemann problem at t = 4 s: (

**a**) contours of water surface elevation; (

**b**) vector field of flow velocities.

D (cm) | d (cm) | h (cm) | B (cm) | b (cm) | Q (m^{3}/s) | S_{w} × 10^{3} | Fr |
---|---|---|---|---|---|---|---|

11.28 | 1.52 | 9.75 | 71.1 | 50.8 | 0.027 | 0.45 | 0.37 |

A (m) | B (m) | C (m) | D (m) | E (m) | F (m) | G (m) | M (m) | N (m) | T (m) | W (m) |
---|---|---|---|---|---|---|---|---|---|---|

0.389 | 0.381 | 0.305 | 0.305 | 0.267 | 0.152 | 0.483 | 2.54 | 0.086 | 0.889 | 0.152 |

Region (#) | Coordinates (m) | Water Depth (m) | u (m/s) | v (m/s) |
---|---|---|---|---|

1 | x ≤ 100, y ≤ 100 | 1 | 10 | 10 |

2 | x > 100, y ≤ 100 | 1 | 0 | 10 |

3 | x ≤ 100, y > 100 | 1 | 10 | 0 |

4 | x > 100, y > 100 | 10 | 0 | 0 |

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**MDPI and ACS Style**

Xu, H.; Liu, X.; Li, F.; Huang, S.; Liu, C.
A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids. *Water* **2018**, *10*, 524.
https://doi.org/10.3390/w10040524

**AMA Style**

Xu H, Liu X, Li F, Huang S, Liu C.
A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids. *Water*. 2018; 10(4):524.
https://doi.org/10.3390/w10040524

**Chicago/Turabian Style**

Xu, Haiyong, Xingnian Liu, Fujian Li, Sheng Huang, and Chao Liu.
2018. "A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids" *Water* 10, no. 4: 524.
https://doi.org/10.3390/w10040524