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ARTICLE

A Study of Traveling Wave Structures and Numerical Investigation of Two-Dimensional Riemann Problems with Their Stability and Accuracy

Abdulghani Ragaa Alharbi^*

Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, 42353, Saudi Arabia

* Corresponding Authors: Abdulghani Ragaa Alharbi. Email: ;

(This article belongs to the Special Issue: Advanced Computational Methods in Fluid Mechanics and Heat Transfer)

Computer Modeling in Engineering & Sciences 2023, 134(3), 2193-2209. https://doi.org/10.32604/cmes.2022.018445

Received 25 November 2021; Accepted 11 May 2022; Issue published 20 September 2022

Abstract

The Riemann wave system has a fundamental role in describing waves in various nonlinear natural phenomena, for instance, tsunamis in the oceans. This paper focuses on executing the generalized exponential rational function approach and some numerical methods to obtain a distinct range of traveling wave structures and numerical results of the two-dimensional Riemann problems. The stability of obtained traveling wave solutions is analyzed by satisfying the constraint conditions of the Hamiltonian system. Numerical simulations are investigated via the finite difference method to verify the accuracy of the obtained results. To extract the approximation solutions to the underlying problem, some ODE solvers in FORTRAN software are applied, and outcomes are shown graphically. The stability and accuracy of the numerical schemes using Fourier’s stability method and error analysis, respectively, to increase the reassurance are investigated. A comparison between the analytical and numerical results is obtained and graphically provided. The proposed methods are effective and practical to be applied for solving more partial differential equations (PDEs).

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