# MacCormack method for solving one-dimensional bed-load sediment transport model

## Authors

• Irfan Said Universitas Hasanuddin
• Agustinus Ribal
• Khaeruddin Khaeruddin

## Keywords:

Bed-load sediment transport model, Shallow water equation, Exner equation, MacCormack Method

## Abstract

In this work, we investigate the numerical solution of one-dimensional bed-load sediment transport model using two steps finite difference method which so-called MacCormack method. Bed-load sediment transport model is composed by the shallow water equation and Exner equation. The Meyer-Peter and Muller (MPM) formula and Wu formula will be used to determine the Grass factor of the bed-load sediment transport. These governing equations will be discretized into predictor and corrector steps of the MacCormack method. The numerical results of the MacCormack method will be validated with an analytical solution of the bed-load sediment transport model. In addition, the MacCormack solution will also be compared with experimental solutions and another numerical method solutions that have existed previously. The numerical results based on MacCormack method give excellent results in which the numerical and the analytical results are hardly differentiated with RMSE of around 00042  or 4,2 .

## References

Berthon, C., Cordier, O., & Le, M. 2012. An analytical solution of the shallow water system [1] Berthon, C., Cordier, O., & Le, M., 2012. An analytical solution of the shallow water system coupled to the Exner Equation. Comptes Rendus Mathematique, Elsevier Masson, 183-186.

Castro Diaz, M., Fernandez-Nieto, E., & Ferreiro, A., 2008. Sediment transport models in shallow water equations and numerical approach by high order finite volume methods. Computer & Fluids, 299-316.

Erami, F. E., & Firoozjaee, A. R., 2019. Numerical solution of bed load transport equations using dicrete least squares meshless (DLSM) method. Applied Mathematical Modelling. 1095-1109.

Firoozjaee, A.R., & Sahebdel, M., 2017. Element-free Galerkin method for numerical simulation of sediment transport equations on regular and irregular distribution of nodes. Engineering Analysis with Boundary Elements. 108-116.

Gunawan, P., & Lhebrard, X., 2015. Hydrostatic relaxation scheme for the 1d shallow water - exner equations in bedload transport. Computers & Fluids, 44-50.

Hoffmann, K., & Chiang, S., 2000. Computational fluid dynamics (Vol. I). Wichita, Kansas: Engineering Education System.

Hudson, J., & Sweby, P., 2003. Formulations for numerically approximating hyperbolic systems governing sediment transport. Journal of Scientific Computing, 225-252.

Juez, C., Murillo, J., & García-Navarro, P., 2014. A 2D weakly-coupled and efficient numerical model for transient shallow flow and movable bed. Advances in Water Resources, 93-109.

Liu, X., & Beljadid, A., 2017. A coupled numerical model for water flow, sediment transport and bed erosion. Computers & Fluids, 273-284.

MacCormack, R., 1969. the effect of viscosity in hypervelocity impact cratering. AIAA Hypervelocity Impact Conference, 69-354.

Magdalena, I., & Pebriansyah, M., 2022. Numerical treatment of finite difference method for solving dam break model on a wet-dry bed withan obsatcle. Results in Engineering, 100382.

Martinez-Aranda, S., Murillo, J. & Garcia-Navarro, P., 2019. A comparative analysis of capacity and non-capacity formulations for the simulation of unsteady flows over finite-depth erodible beds. Advances in Water Resources. 91-112.

Martinez-Aranda, S., Murillo, J., & Garcia-Navarro, P., 2019. A 1D numerical model for the simulation of unsteady and highly erosive flows in rivers. Computers & Fluids , 8-13.

Martinez-Aranda, S., Murillo, J., & Garcia-Navarro, P., 2021. Comparison of new efficient 2D models for the simulation of bedload transport using the augmented Roe approach. Advances in Water Resources, 103931.

Said, I., Ribal, A., & Mahie, A., 2017. Numerical investigation of long-wave breaking on even bottom with linear friction. Far East Journal of Mathematical Sciences, 595-606.

Spinewine, B., & Zech, Y., 2007. Small-scle laboratory dam break waves on movable beds. Journal of Hydraulic Research, 73-86.

Wu, W., 2007. Computational river dynamics. London: Taylor & Francis.

Zhang, S., & Duan, J., 2011. 1D finite volume model of unsteady flow over mobile bed. Journal of Hydrology, 57-68.

2023-01-05

## How to Cite

Said, I., Ribal, A. ., & Khaeruddin, K. (2023). MacCormack method for solving one-dimensional bed-load sediment transport model. Jurnal Matematika, Statistika Dan Komputasi, 19(2), 412-422. https://doi.org/10.20956/j.v19i2.24182

## Section

Research Articles