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

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

* Corresponding Authors: Abdulghani Ragaa Alharbi. Email: ;

*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).## Keywords

Diverse phenomena in nature and technology are described using nonlinear evolution equations (NLEEs). In physics, for instance, the heat transfer and the traveling wave phenomena are successfully modeled by PDEs. In chemistry, the dispersion of a chemically reactive substance is controlled by PDEs. Also, PDEs are invoked to characterize population growth problems. Moreover, most physical incidents of shallow-water waves, quantum mechanics, plasma physics, electricity, fluid dynamics, and more others are investigated using PDEs. In order to better comprehend the qualitative characteristics of such equations, one can study their analytical solutions. A distinct range of traveling wave structures allows us to explain the mechanisms of immense complicated phenomena. As a result, some researchers have revealed numerous effective methods. Some of the powerful approaches are the Sine-Gordon expansion approach [1], the modified simple equation technique [2], the tanh-sech process [3,4], the trial equation method [5], the

The Riemann wave system [16,17] is given by

where

Since our knowledge of constructing exact solutions for the Riemann wave equations is basically based on few techniques, we utilize the generalized exponential rational function approach [8] to construct the exact solutions of the considered equations. This technique depends on Jacobi elliptic functions. As a result, various solitary traveling wave solutions can be simply generated in terms of trigonometric and hyperbolic functions. Regarding the numerical solution, some numerical simulations are presented using the finite difference method to emphasize that the obtained results are accurate. The numerical solutions of the considered system are obtained by approaching the equations on a mesh using the finite-difference notations. The domain is divided into a limited set of grids to achieve meshes for both independent variables x and y. It is well known that the wave of the solutions has areas with rapid spatial changes, for instance, steep fronts structures. In order to resolve these types of areas, fine numbers of grids, for x and y, are required. The step size,

The generalized exponential rational function approach is comprehensively summarized in this section, as presented in [8]. Assume that

is a given nonlinear PDE in the unknown functions

Then, Eq. (2) is converted into

where

where

The balance principle is used to determine the value of N appearing in Eq. (6). Moreover, the coefficients

In this section, we study the exact solutions of system (1) using the generalized exponential rational function approach. Substitute Eq. (3) into system (1) to have

We now integrate each equation in system (7) with respect to

where

where

Family 1: For

Substituting Eq. (10) into the first equation of system (9) and then we insert the result into the first equation of system (8) give an algebraic system whose solutions are given as follows:

Case 1:

Substituting Eq. (11) into system (9) and Eq. (10) gives the exact solutions of system (1) which are illustrated as follows:

Case 2:

Inserting Eq. (13) into system (9) and Eq. (10) yields the exact solutions of system (1) which are expressed as

Case 3:

Plugging Eq. (15) into system (9) and Eq. (10) gives the traveling wave solutions of system (1) which are given by

Case 4:

Putting Eq. (11) into system (9) and Eq. (10) leads to the exact solutions of system (1) which are shown as

Family 2: For

Case 1:

Substituting Eq. (20) into system (9) and Eq. (19) gives the exact solutions of system (1) which are shows as follows:

Case 2:

Inserting Eq. (22) into system (9) and Eq. (19) yields the traveling wave solutions of system (1) which are

Case 3:

Putting Eq. (24) into system (9) and Eq. (19) gives the exact solutions of system (1) which are

Case 4:

Substituting Eq. (26) into system (9) and Eq. (19) leads to the exact solutions of system (1) which are expressed as

4 Stability of the Analytical Solution

We introduce the Hamiltonian system in this section. Hamiltonian system is applied on analytical solutions to test their stability on a specific interval. The Hamiltonian system [24,25] is expressed by

where

When we apply Eqs. (28) and (29) on Eq. (18) over the rectangular domain

where the parameters

5 Finite Difference Semi-Discretization Scheme on a Fixed Mesh

This section is devoted to study the numerical solutions of system (1) over the physical domain

Here,

where

We compute the boundary conditions as follows:

These boundary conditions allow us to employ the fictitious points in computing the spatial derivatives at the boundaries of the domain. It is worth noting that the initial conditions are established by evaluating the exact solution at

where

In order to extract the numerical solutions of the considered equation, we implement the finite difference approach which depends on a standard ODE solver in FORTRAN software, DASPK solver [25]. The standard backward differentiation operators are invoked to approximate the time derivatives in this solver. In addition, the Jacobian matrix of the linearised system is approximated by applying LU factorization. To have less bandwidth for this matrix, we use a unique system numbering for the unknowns

6 Stability of the Numerical Solution

This section investigates the stability of the numerical solution using a Fourier’s stability technique. From the second and third equations of system (8), we have

Substituting these equations into the first equation of system (1) yields

where

where

The boundary conditions are ignored. Consider the point

plugging Eq. (36) into scheme (35) yields

Dividing both sides of the result by

Assume that

Then, Eq. (38) becomes

Hence,

According to the Fourier stability, the stability of the considered scheme occurs if the absolute value of

Taylor series is used in this section to examine the order of the accuracy of scheme (31). We evaluate the truncation error to obtain the order. Suppose that

where

where

Hence,

The leading terms mentioned above are known as the fundamental part of the local truncation error, and we have accepted the truth that

The truncation error, which is generated in every step, is given by

8 Convergence of the Numerical Schemes

Now consider that a sequence of computations is carried out using given initial data, with the refinement of three meshes, so that

We have established above that the implicit schema is unconditional stability. So, we will show that the implicit schemes are unconditional convergence. Suppose that the error e is given by

Now,

Hence by using the fact that

where

Substituting Eq. (46) into Eq. (45) yields

Since the above inequality holds for each

Since the given initial data is used we can identify

But

Hence, the scheme (Eq. (31)) is convergent as

In this section, we discuss the results shown in this work. We extract a distinct range of traveling wave structures of the two-dimensional Riemann problems via the generalized exponential rational function method. The obtained solutions are presented in terms of trigonometric and hyperbolic functions. We examine the stability of Eq. (18) over the rectangular domain

The numerical solutions are discussed by employing the finite difference method to convert the underlying problems into the system of ODEs with keeping time derivatives continuous. Then, I solve the resulted ODEs system using the DASPK solver. This method gives reliable and powerful results. This can be clearly seen in the graphical comparisons presented in the above-mentioned figures. For instance, Figs. 1 and 2 illustrate the behavior of the analytical and numerical solutions for

Table 1 illustrates

We have favorably implemented an accurate finite difference method on a uniform mesh and the generalized exponential rational function method for the two-dimensional Riemann problems. The main advantage of the results is to show the traveling wave structures and prove their accuracy using numerical methods. By comparing the exact solutions using the generalized exponential rational function method with those from the numerical scheme, it was said that the numerical results are almost identical to the analytical results. It is well known that the numerical scheme is stable and allows a meaningful reduction in memory requirements. Using a fine mesh in both spatial variables x and y permits us to resolve the wave-like structures. We can conclude that the used methods can be efficiently applied to more nonlinear evolution models to construct their exact and numerical solutions.

Funding Statement: The author received no specific funding for this study.

Conflicts of Interest: The author declares that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

Alharbi, A. R. (2023). A Study of Traveling Wave Structures and Numerical Investigation of Two-Dimensional Riemann Problems with Their Stability and Accuracy.*CMES-Computer Modeling in Engineering & Sciences, 134(3)*, 2193–2209.