# Application of Large Time Step TVD High Order Scheme to Shallow Water Equations

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## Abstract

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## 1. Introduction

_{max}Δt/Δx, in which u

_{max}is the maximum flow speed, Δt is the time step, and Δx is a spatial increment in the computational mesh. In the 1980s, LeVeque [8,9] proposed the large time step (LTS) method, which led to considerable gains in computational efficiency. In practice, LTS does not conflict with CFL theory in that the interface flux is computed across multiple cells in LTS rather than a single cell in traditional schemes, thus enabling the time step to be increased accordingly. In LTS, more cells take part directly in the computation of fluxes, unlike higher-order schemes that use reconstruction schemes to handle cells adjacent to the interface. Murillo et al. [10] and Morales-Hernández [11,12,13] then applied LTS to solve the SWEs for free surface flows, and similarly, Qian [14] used LTS in solving the Euler equations in aerodynamics. All the foregoing LTS schemes are first order. However, as the CFL number becomes large, spurious oscillations appear in the solution, which do not occur in traditional first-order schemes. Harten introduced the total variation diminishing (TVD) concept [15] that could suppress oscillations for second- or higher-order schemes in the traditional form. Harten [16] then proposed a TVD-LTS scheme that was further developed by Qian and Lee [17].

## 2. TVD-LTS Scheme

#### 2.1. TVD-LTS Scheme for Scalar Case

#### 2.2. TVD-LTS Scheme for Shallow Water Equations

**U**is the vector of stage and discharge,

**F**(

**U**) is the vector of horizontal fluxes,

**S**

_{f}is the vector of source terms, x is distance, t is time, h is the local water depth, u is mean flow velocity, g is the acceleration due to gravity, η is the stage (i.e., water free surface elevation above a fixed horizontal datum), and z is the bed elevation above the same datum.

## 3. Results

#### 3.1. Homogeneous SWEs

_{max}, where K is the CFL number, and λ

_{max}is the maximum eigenvalue in Equation (27). The initial time step is 0.01 s, after which its value depends on the instantaneous value of λ

_{max}. Throughout the computations, the local flow depth and velocity are updated using Equation (30), in which the final term is set to zero for the flat bed.

#### 3.1.1. Two Rarefaction Waves in a Horizontal Channel

#### 3.1.2. Two Shockwaves in a Horizontal Channel

#### 3.1.3. One Rarefaction and One Shockwave in a Horizontal Channel

#### 3.2. Non-Homogeneous SWEs

_{max}). The flow depth and velocity are updated using Equation (30), but with the final term no longer zero at the step.

#### 3.2.1. Two Rarefaction Waves in a Channel Containing a Bed Step

#### 3.2.2. Two Shockwaves in a Channel Containing a Bed Step

#### 3.2.3. One Rarefaction and One Shockwave in a Channel Containing a Bed Step

#### 3.2.4. Transcritical Flow at a Fixed Bed Hump

^{3}/s. The downstream outlet water level is prescribed to be 0.33 m. Initially, the water level is 0.33 m (see Figure 7). Figure 8 shows the analytical solution at a steady state. In the numerical model, the channel is divided into a regular grid of 250 cells, each with a spatial increment of x = 0.1 m. The initial time step is prescribed a value of 0.01 s, which alters each time step thereafter, according to ∆t = K ∆x/λ

_{max}.

**U**between two neighboring time steps less than 10

^{−5}) for transcritical flow at a hump according to CFL number. It should be noted that the use of the same algorithm on the above computer could lead to slightly different CPU time evaluations because of the auto-arrangement of the CPU load and the use of random memory blocks. In all the tests considered, the discrepancy is in the range of ±0.2 s. The average run time per computational step is 5.73 × 10

^{−3}s for a CFL number equal to 1. The run time increases to 31.64 × 10

^{3}s when the CFL number is raised to 4. Unlike the first-order LTS scheme, the TVD-LTS accounts for information from all relevant cells even when their contribution is very small, thus complicating the coefficient (Equation (15)) and leading to an abrupt increase in calculation effort.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Free surface and mass flux profiles for two rarefaction waves in a channel containing a bed step at a given time instant. Rosatti and Begnudelli [23]’s analytical solution (solid lines), well-balanced scheme [22] (red circles), and TVD-LTS predictions (open circles) for CFL = 1 to 10. The solid rectangle indicates the bed shelf.

**Figure 5.**Free surface and mass flux profiles for two shocks in a channel containing a bed step at a given time instant. Rosatti and Begnudelli [23]’s analytical solution (solid lines), well-balanced scheme [22] (red circles), and TVD-LTS predictions (open circles) for CFL = 1 to 10. The solid rectangle indicates the bed shelf.

**Figure 6.**Free surface and mass flux profiles for a single rarefaction and a single shock in a channel containing a bed step at a given time instant. Rosatti and Begnudelli [23]’s analytical solution (solid lines), well-balanced scheme [22] (red circles), and TVD-LTS predictions (open circles) for CFL = 1 to 10. The solid rectangle indicates the bed shelf.

**Figure 7.**Bed elevation and initial water free surface distributions along a channel for the case of transcritical flow over a bed hump.

**Figure 8.**Analytical steady-state free surface elevation distribution [24] along a channel for the case of transcritical flow over a bed hump.

**Table 1.**Initial conditions for a homogeneous case of two rarefaction waves in a frictionless, horizontal channel.

h (m) | q (m^{2}/s) | |
---|---|---|

Left side (x < 12.5 m) | 8.0 | −16.0 |

Right side (x ≥ 12.5 m) | 5.0 | 35.852 |

**Table 2.**Initial conditions for a homogeneous case of two shockwaves in a frictionless, horizontal channel.

h (m) | q (m^{2}/s) | |
---|---|---|

Left side (x < 12.5 m) | 4.0 | 23 |

Right side (x ≥ 12.5 m) | 1.0838 | −5.6199 |

**Table 3.**Initial conditions for a homogeneous case of one rarefaction and one hydraulic bore in a frictionless, horizontal channel.

h (m) | q (m^{2}/s) | |
---|---|---|

Left side (x < 12.5 m) | 5.0 | 0 |

Right side (x ≥ 12.5 m) | 2.0 | 0 |

h (m) | q (m^{2}/s) | z (m) | |
---|---|---|---|

Left side (x < 12.5 m) | 8.0 | −16.0 | 0 |

Right side (x ≥ 12.5 m) | 5.0 | 35.852 | 1 |

h (m) | q (m^{2}/s) | z (m) | |
---|---|---|---|

Left side | 4.0 | 19 | 0 |

Right side | 1.0838 | −2.3685 | 1 |

h (m) | q (m^{2}/s) | z (m) | |
---|---|---|---|

Left side | 5.0 | 0 | 0 |

Right side | 1.0 | 0 | 1 |

**Table 7.**Computational effort by a Godunov-type LTS solver compared to Roe’s scheme for different CFL numbers.

CFL Number | 1 | 2 | 3 | 4 | Roe |
---|---|---|---|---|---|

Steps | 4338 | 2851 | 2338 | 1819 | 6099 |

CPU Time (s) | 24.86 | 32.54 | 46.63 | 57.55 | 3.01 |

Time/Step (10^{−3} s) | 5.73 | 11.41 | 19.94 | 31.64 | 0.49 |

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**MDPI and ACS Style**

Xu, R.; Borthwick, A.G.L.; Xu, B.
Application of Large Time Step TVD High Order Scheme to Shallow Water Equations. *Atmosphere* **2022**, *13*, 1856.
https://doi.org/10.3390/atmos13111856

**AMA Style**

Xu R, Borthwick AGL, Xu B.
Application of Large Time Step TVD High Order Scheme to Shallow Water Equations. *Atmosphere*. 2022; 13(11):1856.
https://doi.org/10.3390/atmos13111856

**Chicago/Turabian Style**

Xu, Renyi, Alistair G. L. Borthwick, and Bo Xu.
2022. "Application of Large Time Step TVD High Order Scheme to Shallow Water Equations" *Atmosphere* 13, no. 11: 1856.
https://doi.org/10.3390/atmos13111856