Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows

Myong Chol Ri; Il Jang

doi:doi:10.11648/j.ajmie.20251005.12

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Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows

Myong Chol Ri^*

, Il Jang

Published in American Journal of Mechanical and Industrial Engineering (Volume 10, Issue 5)

Received: 12 August 2025 Accepted: 9 September 2025 Published: 26 November 2025

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Abstract

Numerical simulations of shallow water flows are widely used to predict global flows such as flood flows in the sea, river and reservoir flows, especially floods due to heavy rainfall and dam break. In particular, reducing the simulation time by applying the Local Time Stepping (LTS) scheme is one way to improve the practical efficiency of numerical simulation. In this paper, we proposed LTS scheme using a structured grid for the numerical simulation of shallow water system. When modeling any terrain, rectangular grid cells are used to facilitate grid generation. To estimate the momentum flux at the grid cell boundaries, we applied the second-order spatial accuracy Godunov finite volume algorithm with Roe approximation solver using the MUSCL method. The LTS scheme is applied to shallow water flow problems with tsunami reflection pattern and its accuracy and efficiency are compared with the traditional global time step (GTS) method. Results show that, with no loss of accuracy, the new LTS algorithm achieves 59-67% CPU time reduction when compared to the GTS method. The proposed LTS scheme accurately reflects time varying water regimes and reflected waves in shallow water flows and can be used for numerical simulations of the three-dimensional shallow water flows with arbitrary topography to reduce the simulation time significantly.

Published in	American Journal of Mechanical and Industrial Engineering (Volume 10, Issue 5)
DOI	10.11648/j.ajmie.20251005.12
Page(s)	96-100
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Shallow Water, Godunov Finite Volume Method, Local Time Stepping

1. Introduction

Nowadays, numerical simulations are widely used to study shallow water flow. The explicit scheme for numerical simulation the shallow water system requires that the time step

Δt

satisfies the CFL condition. For example, in the case of a one-dimensional shallow water problem, this condition is expressed as follows.

α = \frac{aΔt}{Δx} < 1

(1)

Where,

α

is Courant number,

Δx

is grid cell size,

a = |u| + \sqrt{g h}

is the maximum wave velocity in shallow water flow, u is x direction component of the depth-averaged horizontal velocity, ‘g’ is gravitational acceleration.

When the numerical solution is updated with the global time step (GTS)

Δt

, it is constrained by the maximum value of

a / Δx

in the grid. It’s maximum value is determined by the global largest

a

and smallest

Δx

. Since the Courant number

α

in many cells can be much smaller than 1, size of the grid cells is controlled by the parameters to improve the simulation efficiency. However, remeshing locally at each time step is not an efficient approach in numerical simulation, since it rather leads to computational complexity and requires many time steps

[1]	Amanda J. Crossley and Nigel G. Wright, “Local Time Stepping for Modeling Open Channel Flows”, Journal of Hydraulic Research, 2003.129: 455-462. https://doi.org/10.1061/(ASCE)0733-9429(2003)129:6(455) View Article
[2]	Sara Minisini and Elena Zhebel, “Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media”, GEOPHYSICS, VOL. 78, NO. 3.
[3]	Corey J. Trahan and Clint Dawson, “Local time-stepping in Runge-Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations”, Comput. Methods Appl. Mech. Engrg. 217-220 (2012) 139-152. https://doi.org/10.1016/j.cma.2012.01.002 View Article
[4]	Julien Diaz and Marcus J. Grote, “Multi-level explicit local time-stepping methods for second-order wave equations”, Comput. Methods Appl. Mech. Engrg. 291 (2015) 240-265. https://doi.org/10.1016/j.cma.2015.03.02 View Article
[5]	Ben R. Hodges., “A new approach to the local time stepping problem for scalar transport”, Ocean Modelling 77 (2014) 1-19. https://doi.org/10.1016/j.ocemod.2014.02.007 View Article
[6]	Farzam Safarzadeh Maleki and Abdul A. Khan, “A novel Local Time Stepping algorithm for shallow water flow simulation in the discontinuous Galerkin framework”, Applied Mathematical Modelling 40 (2016) 70-84. https://doi.org/10.1016/j.apm.2015.04.038 View Article
[7]	MARCUS J. GROTE and MICHAELA MEHLIN, “CONVERGENCE ANALYSIS OF ENERGY CONSERVING EXPLICIT LOCAL TIME-STEPPING METHODS FOR THE WAVE EQUATION”, Industrial and Applied Mathematics, Vol. 56, No. 2, pp. 994-1021. https://doi.org/10.1137/17M1121925 View Article
[8]	G. Jeanmasson and I. Mary, “On some explicit local time stepping finite volume schemes for CFD”, Journal of Computational Physics, 2019. https://doi.org/10.1016/j.jcp.2019.07.017 View Article
[11]	Jeremy R. Lilly and Giacomo Capodaglio, “Storm Surge Modeling as an Application of Local Time-stepping in MPAS-Ocean”, Journal of Advances in Modeling Earth Systems (2022). https://doi.org/10.5281/zenodo.6908349 View Article
[14]	Xiyan Yang and Shanghong Zhang, “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 View Article
[15]	WEI LENG and ZHU WANG, “High order explicit local time stepping methods for hyperbolic conservation laws”, MATHEMATICS OF COMPUTATION (2020). https://doi.org/10.1090/mcom/3507 View Article

[1-8, 11, 14, 15]

. In

[1]

Amanda J. Crossley and Nigel G. Wright applied the LTS scheme to the numerical simulation of a system of shallow water equations by the triangular grid to evaluate the size of simulation time reduction and the accuracy of the results.

The LTS scheme takes a spatially variable time step which is selected according to a local CFL condition. Compared to the GTS method, the simulation efficiency increases because the cells with the Courant number much smaller than 1 are updated over an integer power of the global time step. A popular LTS scheme involves an hierarchy of L levels of cells where all cells on the same level, ℓ = 1, L use a common time step that is a power-of-two multiple of the base time step, i.e.,

2^{l - 1} Δt

[12, 13, 16]

[9]

J. Rafael Cavalcanti and Michael Dumbser proposed an unstructured grid LTS scheme using the MUSCL-Hancock method to obtain second order of accuracy in space and time. This scheme has been applied to shallow water flows with complex wet and dry areas, confirmed mass conservation and observed improved computational efficiency compared to the standard second-order TVD scheme for scalar transport with global time step (GTS).

[10]

Walter Boscheri and Michael Dumbser proposed a cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular grids that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). The unstructured Lagrangian LTS scheme was applied to typical tests such as the shock-wave tube problem, the cylindrical explosion problem and it was verified that the computational efficiency was improved.

The aim of this paper is to find out the level number L of the hierarchy, the simulation time, and the correctness of the results when applying the LTS scheme to the numerical simulation of a shallow water equation system by the rectangular grid finite volume method.

2. Construction of the Local Time Stepping Scheme

The integral form of a two-dimensional conservative shallow water system neglecting bed slope and bed friction is expressed as follows.

\frac{\partial}{\partial t} \int_{Ω} UdV + \int_{Ω} (\frac{\partial F}{\partial x} + \frac{\partial G}{\partial y}) dV = 0

(2)

U = (\begin{matrix} h \\ hu \\ hv \end{matrix}), F = (\begin{matrix} hu \\ h u^{2} + g h^{2} / 2 \\ huv \end{matrix}), G = (\begin{matrix} hv \\ huv \\ h v^{2} + g h^{2} / 2 \end{matrix})

Where, h is depth, (u, v) are x and y direction components of the depth-averaged horizontal velocity, g is gravitational acceleration.

Applying Green's formula to the system of equations (2), we obtain Eq. (3)

\frac{\partial}{\partial t} \int_{Ω} UdV + \int_{Γ} F_{n} dΓ = 0

(3)

F_{n} = F \cdot n_{x} + G \cdot n_{y}

Where Γ is the boundary of the volume Ω and F_n is the flow rate across the boundary Γ. Applying Eq. (3) to a rectangular grid finite volume and time difference with respect to LTS and take the time difference into account LTS, we obtain the following LTS-type finite-volume equation.

U_{i, j}^{n + p} = U_{i, j}^{n} - \frac{pΔt}{Ω_{i, j}} [{({F_{n}}^{n} Γ)}_{i + 1 / 2, j} - {({F_{n}}^{n} Γ)}_{i - 1 / 2, j} + {({F_{n}}^{n} Γ)}_{i, j + 1 / 2} - {({F_{n}}^{n} Γ)}_{i, j - 1 / 2}]

(4)

F_{n}^{n} = \frac{1}{2} (F_{n} (U_{L}^{n}) + F_{n} (U_{R}^{n}) - \hat{R} |\hat{Λ}| Δ {\overset{̑}{V}}_{RL}^{n})

(5)

Where,

\overset{̑}{R}

and

|\hat{Λ}|

are diagonal matrices consisting of the eigenvector matrix of the Roe mean matrix and the absolute values of the eigenvalues, and

Δ {\overset{̑}{V}}_{RL}^{n}

are characteristic variable difference matrices.

F_{n} (U_{L}^{n})

and

F_{n} (U_{R}^{n})

are fluxes calculated using MUSCL reconstructed data on the left and right sides of the cell boundary, respectively, and the label ‘^’ indicates the quantities obtained from the Roe averages by reconstructed data.

The hierarchical LTS equation (4) was implemented with L levels resulting in an update cycle that repeats every

M = 2^{L - 1}

time steps. At each iteration cycle, care must be taken to ensure conservation of flow rate at the two neighboring cell interfaces, as the neighboring cell variables can progress with different time steps. We have indicated the LTS level at each cell and each cell interface of a rectangular gird at the first step of each cycle, similar to the LTS method applied to the triangular grid finite volume method in

[1]

. The assignment of LTS levels to all cells, faces and vertices follows using a three-step process illustrated in Figure 1.

On the first time step of each LTS cycle, wave speeds

a = |u| + \sqrt{gh}

is computed over interfaces of all cells. Then, applying formula (6) to the four boundaries of the cell, preliminary cell-based LTS level

l_{ci, j}

is assigned to each cell (i, j) according to the local Courant number

α_{i, j}

obtained as the maximum in the cell.

α_{i, j} = \frac{Δt}{Ω_{i, j}} \max_{k = 1, 2, 3, 4} {(a_{k} Γ_{k})}_{i, j}

(6)

l_{ci, j}

is given by the l from the range 1,..., L which satisfies the following inequality,

α_{0} / 2^{l} < α_{i, j} < α_{0} / 2^{l - 1}

, with the exception of level 1 which is controlled by

α_{i, j} < α_{0} / 2

and level L which is controlled by

α_{i, j} < α_{0} / 2^{L - 1}

. The parameter

α_{0}

represents a target Courant number, and by trial and error it was set to 0.8 or 20% below the theoretical stability limit. A value less than unity was selected because

l_{ci, j}

is assigned once every M time steps, and

α_{i, j}

may increase during this sequence. Hence,

α_{0} = 0.8

provides a degree of resilience against rapidly changing flow conditions that may cause instability though it cannot completely preclude this possibility.

In step 2, LTS level of interfaces at each cell,

l_{fi, jK}

is assigned as the minimum of the two neighbouring cell-based LTS levels.

In step 3, the final value of the LTS level of interfaces at each cell is given as the minimum of the four neighbouring face-based LTS levels.

Download: Download full-size image

Figure 1. LTS level setting of cell and cell interfaces.

①,②,③- cell level, 1, 2, 3- level of cell interfaces

Once LTS levels are assigned, the model carries out a complete LTS cycle of M time loops, i.e., m = 1,..., M. During each sweep, data are only operated on when the LTS level is less than or equal to a threshold

l_{0}

. The selection of

l_{0}

depends on whether m is odd or even. If m is odd,

l_{0} = 1

. If m is even,

l_{0}

further depends on m. For example, the values of

l_{0}

correspond to L = 3, m = 1,..., 4 are 1, 2, 1, 3, and the values of

l_{0}

correspond to L = 4, m = 1,..., 8 are 1, 2, 1, 3, 1, 2, 1, 4.

Download: Download full-size image

Figure 2. Computational domain and initial water surface.

3. Numerical Example

The numerical experiment is conducted in a square region with a length of 50 m and a width of 50 m surrounded by a vertical wall boundary. Figure 2 shows the computational domain and initial water level. At the initial moment, the bottom is filled with water at a depth of 0.1 m, and a water column with a height of 1 m is placed in right corner. We simulated the flow in which this water column collapses. The basic time step is taken as

Δt = 0.15 s

Figure 3 shows the development of the water column with time. We presented simulation results for hierarchical structures of L from 1 to 5. Many numerical experiments show that the LTS schemes for hierarchical structures above L = 6 diverge and the numerical fails. As shown in Figure 3, the initial water column gradually collapses over time, and at time = 18 s, a reflected wave is formed by colliding with the opposite side. This reflected wave propagates upstream again and begins to interfere with each other. At time = 30 s, the highest water column is formed on the opposite side of the initial water column, and the water column collapses and the movement of the water surface takes place repeatedly. Through simulation, it can be seen that the LTS scheme accurately reflects the propagation process of the water surface without divergence during the numerical simulation.

Download: Download full-size image

Figure 3. The development of water column change with time.

The numerical results for hierarchies L from 1 to 5 are in good agreement. Figure 4 shows the free surface height along the x-axis at the y = 25 m section at time= 30 s for each of the hierarchical structures. As shown in Figure 4, it is concluded that the differences of results are almost negligible.

Download: Download full-size image

Figure 4. water level variation along the x-axis in the y = 25 m section at time= 30 s.

However, as shown in Table 1, the difference in computation time is noticeable. For the results with the same accuracy, the LTS scheme with L = 2 reduces the computation time by 67% compared to the GTS scheme and the computation time by 58.9% with L = 5, respectively. The rate of decrease of computational time decreases rapidly as L increases.

Table 1. comparison of CPU computation time with level.

L	CPU computation time, s	Computation time percent, %
1	425	100
2	285	67
3	271	63.7
4	256	60.2
5	249	58.9

4. Conclusions

In the paper, we investigated the relationship between the number of setting level L and CPU computation time when applying the LTS scheme to the numerical simulation of a homogeneous shallow water by the rectangular grid finite volume method. As setting level L increases, the simulation time decreases significantly and there is little change in the accuracy of the simulation results. However, there is a limit in the set value of L, which diverges beyond that threshold, causing simulation failure. The limit is 5 in our computational experiments.

Abbreviations

MUSCL	Monotone Upwind Scheme for Conservation Laws
CFL	Courant-Friedrichs-Lewy
ALE	Arbitrary-Lagrangian-Eulerian

Acknowledgments

It is also the result of collaborative research with State Academy of Sciences.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Funding

This work was partially supported by State Academy of Sciences.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Amanda J. Crossley and Nigel G. Wright, “Local Time Stepping for Modeling Open Channel Flows”, Journal of Hydraulic Research, 2003.129: 455-462. https://doi.org/10.1061/(ASCE)0733-9429(2003)129:6(455)
[2]	Sara Minisini and Elena Zhebel, “Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media”, GEOPHYSICS, VOL. 78, NO. 3.
[3]	Corey J. Trahan and Clint Dawson, “Local time-stepping in Runge-Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations”, Comput. Methods Appl. Mech. Engrg. 217-220 (2012) 139-152. https://doi.org/10.1016/j.cma.2012.01.002
[4]	Julien Diaz and Marcus J. Grote, “Multi-level explicit local time-stepping methods for second-order wave equations”, Comput. Methods Appl. Mech. Engrg. 291 (2015) 240-265. https://doi.org/10.1016/j.cma.2015.03.02
[5]	Ben R. Hodges., “A new approach to the local time stepping problem for scalar transport”, Ocean Modelling 77 (2014) 1-19. https://doi.org/10.1016/j.ocemod.2014.02.007
[6]	Farzam Safarzadeh Maleki and Abdul A. Khan, “A novel Local Time Stepping algorithm for shallow water flow simulation in the discontinuous Galerkin framework”, Applied Mathematical Modelling 40 (2016) 70-84. https://doi.org/10.1016/j.apm.2015.04.038
[7]	MARCUS J. GROTE and MICHAELA MEHLIN, “CONVERGENCE ANALYSIS OF ENERGY CONSERVING EXPLICIT LOCAL TIME-STEPPING METHODS FOR THE WAVE EQUATION”, Industrial and Applied Mathematics, Vol. 56, No. 2, pp. 994-1021. https://doi.org/10.1137/17M1121925
[8]	G. Jeanmasson and I. Mary, “On some explicit local time stepping finite volume schemes for CFD”, Journal of Computational Physics, 2019. https://doi.org/10.1016/j.jcp.2019.07.017
[9]	J. Rafael Cavalcanti and Michael Dumbser, “A Conservative Finite Volume Scheme with Time-Accurate Local Time Stepping for Scalar Transport on Unstructured Grids”, Advances in Water Resources, 2015. https://doi.org/10.1016/j.advwatres.2015.10.002
[10]	Walter Boscheri and Michael Dumbser, “High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes”, Journal of Computational Physics 291 (2015) 120-150. https://doi.org/10.1016/j.jcp.2015.02.052
[11]	Jeremy R. Lilly and Giacomo Capodaglio, “Storm Surge Modeling as an Application of Local Time-stepping in MPAS-Ocean”, Journal of Advances in Modeling Earth Systems (2022). https://doi.org/10.5281/zenodo.6908349
[12]	Shenghan Hu and Mengyao Zhang, “Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space-Time Coupled Generalized Finite Difference Scheme”, Applied sciences (2023). https://doi.org/10.3390/app13148504
[13]	Diego Fernando and Arlex Chaves, “Decoupled solution of the sediment transport and 2D shallow water equations using the finite volume method”, Results in Engineering 15 (2022) 100504. https://doi.org/10.1016/j.rineng.2022.100504
[14]	Xiyan Yang and Shanghong Zhang, “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
[15]	WEI LENG and ZHU WANG, “High order explicit local time stepping methods for hyperbolic conservation laws”, MATHEMATICS OF COMPUTATION (2020). https://doi.org/10.1090/mcom/3507
[16]	Miguel Masó and Alessandro Franci, “A Lagrangian-Eulerian procedure for the coupled solution of the Navier-Stokes and shallow water equations for landslide-generated waves”, Advanced Modeling and Simulation in Engineering Sciences (2022). https://doi.org/10.1186/s40323-022-00225-9

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APA Style

Ri, M. C., Jang, I. (2025). Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows. American Journal of Mechanical and Industrial Engineering, 10(5), 96-100. https://doi.org/10.11648/j.ajmie.20251005.12

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ACS Style

Ri, M. C.; Jang, I. Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows. Am. J. Mech. Ind. Eng. 2025, 10(5), 96-100. doi: 10.11648/j.ajmie.20251005.12

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AMA Style

Ri MC, Jang I. Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows. Am J Mech Ind Eng. 2025;10(5):96-100. doi: 10.11648/j.ajmie.20251005.12

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@article{10.11648/j.ajmie.20251005.12,
  author = {Myong Chol Ri and Il Jang},
  title = {Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows
},
  journal = {American Journal of Mechanical and Industrial Engineering},
  volume = {10},
  number = {5},
  pages = {96-100},
  doi = {10.11648/j.ajmie.20251005.12},
  url = {https://doi.org/10.11648/j.ajmie.20251005.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmie.20251005.12},
  abstract = {Numerical simulations of shallow water flows are widely used to predict global flows such as flood flows in the sea, river and reservoir flows, especially floods due to heavy rainfall and dam break. In particular, reducing the simulation time by applying the Local Time Stepping (LTS) scheme is one way to improve the practical efficiency of numerical simulation. In this paper, we proposed LTS scheme using a structured grid for the numerical simulation of shallow water system. When modeling any terrain, rectangular grid cells are used to facilitate grid generation. To estimate the momentum flux at the grid cell boundaries, we applied the second-order spatial accuracy Godunov finite volume algorithm with Roe approximation solver using the MUSCL method. The LTS scheme is applied to shallow water flow problems with tsunami reflection pattern and its accuracy and efficiency are compared with the traditional global time step (GTS) method. Results show that, with no loss of accuracy, the new LTS algorithm achieves 59-67% CPU time reduction when compared to the GTS method. The proposed LTS scheme accurately reflects time varying water regimes and reflected waves in shallow water flows and can be used for numerical simulations of the three-dimensional shallow water flows with arbitrary topography to reduce the simulation time significantly.
},
 year = {2025}
}

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TY  - JOUR
T1  - Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows

AU  - Myong Chol Ri
AU  - Il Jang
Y1  - 2025/11/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajmie.20251005.12
DO  - 10.11648/j.ajmie.20251005.12
T2  - American Journal of Mechanical and Industrial Engineering
JF  - American Journal of Mechanical and Industrial Engineering
JO  - American Journal of Mechanical and Industrial Engineering
SP  - 96
EP  - 100
PB  - Science Publishing Group
SN  - 2575-6060
UR  - https://doi.org/10.11648/j.ajmie.20251005.12
AB  - Numerical simulations of shallow water flows are widely used to predict global flows such as flood flows in the sea, river and reservoir flows, especially floods due to heavy rainfall and dam break. In particular, reducing the simulation time by applying the Local Time Stepping (LTS) scheme is one way to improve the practical efficiency of numerical simulation. In this paper, we proposed LTS scheme using a structured grid for the numerical simulation of shallow water system. When modeling any terrain, rectangular grid cells are used to facilitate grid generation. To estimate the momentum flux at the grid cell boundaries, we applied the second-order spatial accuracy Godunov finite volume algorithm with Roe approximation solver using the MUSCL method. The LTS scheme is applied to shallow water flow problems with tsunami reflection pattern and its accuracy and efficiency are compared with the traditional global time step (GTS) method. Results show that, with no loss of accuracy, the new LTS algorithm achieves 59-67% CPU time reduction when compared to the GTS method. The proposed LTS scheme accurately reflects time varying water regimes and reflected waves in shallow water flows and can be used for numerical simulations of the three-dimensional shallow water flows with arbitrary topography to reduce the simulation time significantly.

VL  - 10
IS  - 5
ER  -

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Author Information

Myong Chol Ri

Faculty of Physical Engineering, Kim Chaek University of Technology, Pyongyang, DPR Korea

Contact Email

http://orcid.org/0009-0007-5835-904X
Il Jang

Faculty of Physical Engineering, Kim Chaek University of Technology, Pyongyang, DPR Korea

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Table 1

Table 1. comparison of CPU computation time with level.

Plain Text BibTeX RIS

APA Style

Ri, M. C., Jang, I. (2025). Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows. American Journal of Mechanical and Industrial Engineering, 10(5), 96-100. https://doi.org/10.11648/j.ajmie.20251005.12

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ACS Style

Ri, M. C.; Jang, I. Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows. Am. J. Mech. Ind. Eng. 2025, 10(5), 96-100. doi: 10.11648/j.ajmie.20251005.12

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AMA Style

Ri MC, Jang I. Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows. Am J Mech Ind Eng. 2025;10(5):96-100. doi: 10.11648/j.ajmie.20251005.12

Copy | Download

@article{10.11648/j.ajmie.20251005.12,
  author = {Myong Chol Ri and Il Jang},
  title = {Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows
},
  journal = {American Journal of Mechanical and Industrial Engineering},
  volume = {10},
  number = {5},
  pages = {96-100},
  doi = {10.11648/j.ajmie.20251005.12},
  url = {https://doi.org/10.11648/j.ajmie.20251005.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmie.20251005.12},
  abstract = {Numerical simulations of shallow water flows are widely used to predict global flows such as flood flows in the sea, river and reservoir flows, especially floods due to heavy rainfall and dam break. In particular, reducing the simulation time by applying the Local Time Stepping (LTS) scheme is one way to improve the practical efficiency of numerical simulation. In this paper, we proposed LTS scheme using a structured grid for the numerical simulation of shallow water system. When modeling any terrain, rectangular grid cells are used to facilitate grid generation. To estimate the momentum flux at the grid cell boundaries, we applied the second-order spatial accuracy Godunov finite volume algorithm with Roe approximation solver using the MUSCL method. The LTS scheme is applied to shallow water flow problems with tsunami reflection pattern and its accuracy and efficiency are compared with the traditional global time step (GTS) method. Results show that, with no loss of accuracy, the new LTS algorithm achieves 59-67% CPU time reduction when compared to the GTS method. The proposed LTS scheme accurately reflects time varying water regimes and reflected waves in shallow water flows and can be used for numerical simulations of the three-dimensional shallow water flows with arbitrary topography to reduce the simulation time significantly.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Local Time Stepping Scheme Using Structured Grids for Modelling of Shallow Water Flows

AU  - Myong Chol Ri
AU  - Il Jang
Y1  - 2025/11/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajmie.20251005.12
DO  - 10.11648/j.ajmie.20251005.12
T2  - American Journal of Mechanical and Industrial Engineering
JF  - American Journal of Mechanical and Industrial Engineering
JO  - American Journal of Mechanical and Industrial Engineering
SP  - 96
EP  - 100
PB  - Science Publishing Group
SN  - 2575-6060
UR  - https://doi.org/10.11648/j.ajmie.20251005.12
AB  - Numerical simulations of shallow water flows are widely used to predict global flows such as flood flows in the sea, river and reservoir flows, especially floods due to heavy rainfall and dam break. In particular, reducing the simulation time by applying the Local Time Stepping (LTS) scheme is one way to improve the practical efficiency of numerical simulation. In this paper, we proposed LTS scheme using a structured grid for the numerical simulation of shallow water system. When modeling any terrain, rectangular grid cells are used to facilitate grid generation. To estimate the momentum flux at the grid cell boundaries, we applied the second-order spatial accuracy Godunov finite volume algorithm with Roe approximation solver using the MUSCL method. The LTS scheme is applied to shallow water flow problems with tsunami reflection pattern and its accuracy and efficiency are compared with the traditional global time step (GTS) method. Results show that, with no loss of accuracy, the new LTS algorithm achieves 59-67% CPU time reduction when compared to the GTS method. The proposed LTS scheme accurately reflects time varying water regimes and reflected waves in shallow water flows and can be used for numerical simulations of the three-dimensional shallow water flows with arbitrary topography to reduce the simulation time significantly.

VL  - 10
IS  - 5
ER  -

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