Stability analysis and convergence rate of a two-step predictor-corrector approach for shallow water equations with source terms

Rubayyi T. Alqahtani; Jean C. Ntonga; Eric Ngondiep; Rubayyi T. Alqahtani; Jean C. Ntonga; Eric Ngondiep

doi:10.3934/math.2023465

AIMS Mathematics

2023, Volume 8, Issue 4: 9265-9289. doi: 10.3934/math.2023465

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Stability analysis and convergence rate of a two-step predictor-corrector approach for shallow water equations with source terms

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), 90950 Riyadh 11632, Saudi Arabia
2.
Hydrological Research Centre, Institute for Geological and Mining Research, 4110 Yaounde-Cameroon

Received: 24 December 2022 Revised: 24 January 2023 Accepted: 01 February 2023 Published: 14 February 2023
MSC : 70K20, 35F61, 65M12, 93C20, 65M06, 93B18

This paper deals with a two-step explicit predictor-corrector approach so-called the two-step MacCormack formulation, for solving the one-dimensional nonlinear shallow water equations with source terms. The proposed two-step numerical scheme uses the fractional steps procedure to treat the friction slope and to upwind the convection term in order to control the numerical oscillations and stability. The developed scheme uses both forward and backward difference formulations in the predictor and corrector steps, respectively. The linear stability of the constructed technique is deeply analyzed using the Von Neumann stability approach whereas the convergence rate of the proposed method is numerically obtained in the $ L^{2} $-norm. A wide set of numerical examples confirm the theoretical results.
- shallow water equations,
- source terms,
- a two-step explicit predictor-corrector approach,
- Fourier stability analysis,
- linear stability condition,
- convergence rate
Citation: Rubayyi T. Alqahtani, Jean C. Ntonga, Eric Ngondiep. Stability analysis and convergence rate of a two-step predictor-corrector approach for shallow water equations with source terms[J]. AIMS Mathematics, 2023, 8(4): 9265-9289. doi: 10.3934/math.2023465

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Abstract

This paper deals with a two-step explicit predictor-corrector approach so-called the two-step MacCormack formulation, for solving the one-dimensional nonlinear shallow water equations with source terms. The proposed two-step numerical scheme uses the fractional steps procedure to treat the friction slope and to upwind the convection term in order to control the numerical oscillations and stability. The developed scheme uses both forward and backward difference formulations in the predictor and corrector steps, respectively. The linear stability of the constructed technique is deeply analyzed using the Von Neumann stability approach whereas the convergence rate of the proposed method is numerically obtained in the $ L^{2} $-norm. A wide set of numerical examples confirm the theoretical results.

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