THE SIMULATION OF ONE-DIMENSIONAL SHALLOW WATER WAVE EQUATION WITH MACCORMACK SCHEMES

  • Iffah Nurlathifah Fikri Mathematics Study Program, Mathematics Department, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Gadjah Mada
  • Sumardi Sumardi Mathematics Study Program, Mathematics Department, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Gadjah Mada
DOI: https://doi.org/10.30598/barekengvol16iss2pp729-742
Keywords: shallow water wave, maccormack method, topography, pond, tsunami

Abstract

Many practical problems can be modeled using the one-dimensional shallow water wave equation. Therefore, the solution to the one-dimensional shallow water wave equation will be discussed to solve this problem. The research method used was the study of literature related to the shallow water wave equation and its solution method. The one-dimensional shallow water wave equation can be derived from the law of conservation of mass and the law of conservation of momentum. In this study, one of the finite difference methods will be discussed, namely the MacCormack method. The MacCormack method consists of two steps, namely the predictor and corrector steps. The MacCormack method was used to perform numerical simulations of the pond and tsunami models for one-dimensional (1D) shallow water wave equations with flat and non-flat topography. The simulation results showed that the channel's topography could affect the water surface's height and velocity. At the same time, a channel with a non-flat topography had a slower water velocity than the water velocity of a channel with a flat topography.

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References

A. Trujillo and H. Thurman, Essentials of Oceanography, Boston: Pearson, 2016.

M. Jamhuri, "Simulasi Perambatan Tsunami menggunakan Persamaan Gelombang Air Dangkal [Simulation of Tsunami Propagation using the Shallow Water Wave Equation]," Universitas Islam Negeri Maliki, Malang, 2014.

S. Hati, K. Yulianti and R. Marwati, "Simulasi Sistem Persamaan Gelombang Air Dangkal Menggunakan Metode Numeris Lax-Friedrichs [Simulation of Shallow Water Wave Equation System Using Lax-Friedrichs . Numerical Method]," Jurnal Eureka Matika, vol. 8, 2020.

D. Prayitno and Sumardi, "Simulations of shallow water equations by finite difference WENO schemes with multilevel time discretization," AIP Conference Proceedings, vol. 2268, no. 1, 2020.

S. R. Pudjaprasetya and I. Magdalena, " Momentum Conservative Schemes for Shallow Water Flows," East Asian Journal on Applied Mathematics, 2013.

H. P. Gunawan, "Numerical Simulation of Shallow Water Equations and Related Models," HAL, 2015.

R. Setiyowati and Sumardi, "A Simulation of Shallow Water Wave Equation Using Finite Volume Method: Lax-Friedrichs Scheme," Journal of Physics: Conference Series, vol. 1306, 2019.

P. Crowhurst, "Numerical Solution of One Dimentional Shallow Water Equation," Australian Mathematical Sciences Institute, 2013.

I. Wahidah, " Simulasi Numerik Model Sisko Untuk Aliran Darah Tak-Tunak Dalam Pembuluh [Numerical Simulation Sisko Model for Unsteady Blood Flow in Vessels]," Universitas Gadjah Mada, Yogyakarta, 2018.

N. Pochai, "Numerical treatment of a modified MacCormack scheme in a nondimensional form of the water quality models in a nonuniform flow stream," Journal of Applied Mathematics, 2014.

N. L. H. Zendrato, A. Chrysanti, B. P. Yakti, M. B. Adityawan, Widyaningtias and Y. Suryadi, "Application of Finite Difference Schemes to 1D St. Venant for Simulating Weir Overflow," in MATEC Web of Conferences, 2018.

E. Ngondiep, A. Rubayyi and J. C. Ntonga, "A MacCormack Method for Complete Shallow Water Equations with Source Terms," American Mathematical Society, no. 65, 2019.

M. A. Abdelrahman, "On the Shallow Water Equations," Zeitschrift fur Naturforschung a, vol. 72, no. 9, pp. 873-879, 2017.

H. S. Shaw, "Numerical Modelling of Tsunami Wave Equations," National Institute of Technology Rourkela, 2014.

P. R. Pinet, Invitation to Oceanography Seventh Edition, Jones & Bartlett Publisher, 2014.

Published
2022-06-01
How to Cite
[1]
I. Fikri and S. Sumardi, “THE SIMULATION OF ONE-DIMENSIONAL SHALLOW WATER WAVE EQUATION WITH MACCORMACK SCHEMES”, BAREKENG: J. Math. & App., vol. 16, no. 2, pp. 729-742, Jun. 2022.
Issue
Vol 16 No 2 (2022): BAREKENG: Jurnal Ilmu Matematika dan Terapan
Section
Articles

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