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).

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 [

The Riemann wave system [

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 [

The generalized exponential rational function approach is comprehensively summarized in this section, as presented in [

Then,

The balance principle is used to determine the value of

In this section, we study the exact solutions of system

We now integrate each equation in system

Family 1: For

Substituting

Case 1:

Substituting

Case 2:

Inserting

Case 3:

Plugging

Case 4:

Putting

Family 2: For

Case 1:

Substituting

Case 2:

Inserting

Case 3:

Putting

Case 4:

Substituting

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 [

When we apply

This section is devoted to study the numerical solutions of system

Here,

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

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 [

This section investigates the stability of the numerical solution using a Fourier’s stability technique. From the second and third equations of system

Substituting these equations into the first equation of system

The boundary conditions are ignored. Consider the point

Dividing both sides of the result by

Assume that

Then,

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

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

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

Now,

Hence by using the fact that

Substituting

Since the above inequality holds for each

Since the given initial data is used we can identify

But

Hence, the scheme (

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

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,

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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