Galerkin-compact finite difference residual corrections for nonlinear second order wave equations

Abstract

In this study, numerical solutions for second-order nonlinear wave equations were obtained, and then were subsequently refined the numerical solution using residual corrections. The proposed method uniquely combines the Galerkin method with the compact finite difference (CFD) method, representing a novel approach in this field. We initially develop a rigorous formulation of second-order wave equations using the Galerkin weighted residual method and obtain numerical results. To solve these equations, third degree Bernstein polynomials are utilized as basis functions in the trial solution. Then we apply our proposed residual correction scheme to refine the numerical solution where fourth-order CFD method are used to solve the associated error wave equations in compliance with the error boundary, and initial conditions. The improved approximations are obtained by adding error values that were obtained based on the estimates of the error wave equation to the weighted residual values. Here Galerkin method and residual correction come together and produce highly accurate results. We also discuss the stability and convergence analysis of our proposed residual correction scheme. Numerical outcomes and absolute errors are compared with exact solutions and solutions found in published literature numerically for different values of space and time step sizes to verify our proposed residual correction scheme. High precision is obtained in case of residual corrections.

Introduction

A numerical method is a mathematical tool that is used mainly to solve numerical problems. The fundamental advantage of numerical approaches is the ability to obtain numerical solutions that may be analytically intractable to solve with traditional analytical methods. These methods involve algorithms that use numerical calculations to obtain approximate solutions, often by attractively refining an initial guess until a desired level of accuracy is achieved. The history of numerical method is vast and spans centuries. It traces its origin back to ancient civilizations such as Babylonians, Egyptians and Greeks [1, 2]. In the middle age, Ibn-Al-Haytham and Johannes Kepler made significant contribution to interpolation and approximation techniques. Later Newton and Leibniz provides tools for solving differential equations which form the basis of many numerical methods [3]. In nineteenth century, Carl Friedrich Gauss and Pierre Simon Laplace developed numerical integration technique. The twentieth century saw the emergence of Finite Difference Method for solving Differential Equation numerically [3, 4].

The history of partial differential equations (PDEs) is deeply intertwined with the development of mathematics, physics, and engineering as these equations have been crucial in modeling physical phenomena such as heat, sound, fluid flow, and electromagnetism [5]. Wave equation is a hyperbolic PDE which describes the nature of waves that appear frequently in physics including propagation of sound wave, light, water wave, elasticity, fluid dynamics, vibrations etc [6]. Wave equations, both linear and nonlinear, may replicate a vast array of natural processes. They are used in numerous scientific and engineering applications. In the absence of analytical solution, numerical modeling allows us to analyze and explain a phenomenon. D’Alembert derived the wave equation, a second-order linear PDE describing the motion of vibrating strings in 1747 [7]. Wave equations may replicate a vast array of natural processes. An important area of research is the development of numerical methods for solving PDEs [8]. Russian mathematician Boris Galerkin first introduced Galerkin method in the early twentieth century for solving integral equation which was later extended to Differential Equations [8, 9]. Originally it was developed for solving Partial Differential Equation [10, 11]. Another important method for solving PDEs which is compact finite difference method [6, 12]. It traces it’s origin back to the work of various researchers including Gottileb, Husdaini, Orszag who first gave the basic idea of compact finite difference method for solving PDEs [13, 14–15]. Over the years, researchers have contributed to refine and improve more advance compact finite difference schemes, making them applicable to increasingly complex problems [16, 17].

Numerical simulations of nonlinear wave equations, such as the generalized equal-width (GEW) and modified equal-width (MEW) equations, have been extensively studied using various spline-based techniques [18, 19–20]. These studies demonstrate the accuracy and stability of spline-based methods in capturing solitary wave solutions while effectively conserving key physical quantities [18]. In particular, the approach presented in [19] improves computational efficiency without compromising precision, especially for long-time simulations of wave dynamics. Furthermore, the analysis in [20] employs cubic Hermite B-splines in a collocation framework, yielding smooth and accurate solutions; the Hermite basis also provides additional flexibility for handling boundary conditions compared to standard B-splines. Recent advances in numerical methods for solving partial differential equations (PDEs) further support the effectiveness of spline-based techniques, particularly in modeling nonlinear wave and diffusion phenomena. A synthesis of findings from three key studies [21, 22–23] reveals the applicability of Galerkin finite element and collocation methods using cubic B-splines and Hermite B-splines. Specifically, the study in [21] introduces a Galerkin finite element scheme with cubic B-spline basis functions to accurately simulate dissipative and dispersive effects in the Benjamin–Bona–Mahony–Burgers (BBMB) equation, demonstrating strong convergence and stability for shock and solitary wave profiles. The use of a lumped mass matrix in [22] further enhances computational performance while preserving accuracy, with numerical results showing excellent agreement with analytical solutions even over extended time periods. Meanwhile, the work in [23] addresses parabolic PDEs by applying a collocation method with cubic Hermite B-splines to the heat equation, where the continuity of Hermite B-splines enables smooth approximation of temperature distributions and effective treatment of boundary conditions and sharp gradients.

The history of residual correction methods [24, 25] is related to the advancement of mathematical methodologies for solving systems of equations and improving the accuracy of numerical approximations. The pursuit of more precise and effective iterative techniques by numerical analysts led to a formalization of the residual correction idea. Rather than only making estimates based on the residual, mathematicians started creating algorithms that explicitly used different optimization approaches to minimize the residual. Oliveira introduced the residual correction methods for the first time using the explicit finite difference method [24]. Afterwards, Celik examined the use of a truncated Chebyshev series in the corrected collocation method to estimate the eigenvalues of Sturm-Liouville and periodic Sturm–Liouville problems [26]. The numerical and analytical investigation of solitary wave solutions in nonlinear wave equations remains a prominent area of research in applied mathematics and physics. Two notable studies offer valuable insights into distinct aspects of solitary wave behavior, focusing respectively on the Regularized Long Wave (RLW) equation and the quadratic-nonlinear Korteweg–de Vries (KdV) equation [27, 28]. The approach in [27] employs a hybrid method combining finite difference and spectral techniques, achieving high accuracy in simulating wave propagation and interaction over extended spatial domains. This study emphasizes the robustness of the proposed scheme under varied initial and boundary conditions, underscoring its applicability in real-world scenarios such as fluid dynamics and coastal engineering. In contrast, the study in [28] utilizes both analytical and numerical methods to explore the evolution of single and double solitons across short- and long-term time scales. The results illuminate key features of soliton stability and interactions, thereby contributing to a deeper understanding of nonlinear dispersive systems and their solitary wave dynamics.

In our study, we are motivated to solve second order nonlinear wave equations using Galerkin method [29]. Galerkin method involves approximating the solution of a wave equation by seeking a trial solution first which involve basis functions [30, 31, 32–33]. Next we want to reduce the error in the solution obtained from the Galerkin method by applying our proposed residual correction scheme. For residual correction here we have used fourth order compact finite difference method. By employing this method, our primary objective is to enhance the accuracy of the numerical solution. This is achieved by systematically reducing the associated numerical errors obtained from Galerkin method, which in turn leads to a significant improvement in the overall accuracy of the solution to the wave equation [34, 35–36].

The paper is organized as follows: Sect. 1 of this research paper contains an introduction and a review of the literature, while Sect. 2 presents a generalized formulation of the Galerkin weighted residual approach for solving nonlinear wave equations. Section 3 discussed a generalized formulation of the fourth order compact finite difference approach. Our suggested residual correction scheme is covered in Sect. 4, while the stability and convergence analysis of our suggested scheme are covered in Sect. 5. Section 6 displays a number of numerical application of second order nonlinear wave equations. In section 7, we finally report the findings from our research work.

Wave equations: Galerkin method

The Galerkin method involves approximating the solution of a PDE by seeking a trial solution. Consider the standard second order nonlinear wave equation of the form [9]:

2.1

with domain for space and for time along with Fixed Dirichlet boundary conditions: and initial conditions: where is called the diffusion coefficient, v is the function of both space x and time t, f is the reaction-advection function, is called the diffusion term. They are used mainly to describe time-dependent phenomena. If the boundary condition are non-homogeneous, then the trial solution of Eq. (2.1) be of the form:

2.2

where satisfies the whole boundary conditions and satisfies the homogeneous form of the given boundary conditions. Then the weighted residual equation for homogeneous boundary conditions over the interval may be written as:

2.3

Integrating by parts of the second terms and substituting the trial solution Eq. (2.2) in Eq. (2.3), we obtain:

2.4

where,Eq. (2.4) can be expressed as follows in a suitable matrix form:

2.5

The initial condition will now be satisfied in the Galerkin sense.

2.6

and

2.7

Solving the above two Eqs. (2.6) and (2.7) we will obtain the value of which will be used as initial conditions for the above system of nonlinear ODEs. The system of nonlinear ordinary differential equations will be treated analytically or by using several other numerical approaches [1, 3]. Once we get the value of a(t), substituting the value into the trial solution in Eq. (2.2), we will obtain the approximate solution of second order nonlinear wave equations Eq. (2.1).

Wave equations: CFD method

The compact finite difference (CFD) formulation, commonly known as the Hermitian formulation, is a numerical method for calculating finite difference approximations. Here we derive fourth order compact finite difference method for solving second order nonlinear wave equation. Consider the second order nonlinear wave equation Eq. (2.1).

Here both initial and boundary conditions are provided to solve the above equation. To describe the method, the intervals [a, b] and [0, T] are initially partitioned into and equal sub-intervals, resulting in mesh lengths of h and k respectively [6].

Suppose that and be the mesh points where and for and .

For simplicity suppose that

Following the literature [6], it was obtained:

3.1

for ,    and

3.2

Here andThus, the aforementioned finite difference schemes provided in Eqs. (3.1) and (3.2) result in a set of equations [6]. By applying matrix inverse method, it provides a approximate numerical solution to the second order wave equation implicitly.

Residual correction

Relative error is the amount of error that is left over after comparing an approximation to the real value of a solution [37]. The method of residual correction involves solving the error differential equation in order to minimize the residuals. Error differential equations are obtained by taking the difference between actual and approximate differential equation. For solving the error differential equation we will apply fourth order compact finite differnece method. If the exact solution is v(x, t) and the approximate solution is , then residual can be acquired by:Substituting this value into Eq. (2.1) we obtain:

4.1

which is called the error wave equation. Applying the same procedure in initial and boundary conditions we obtain:

4.2

Using the fourth order compact finite difference approach described in Sect. 3, we will solve the error wave Eq. (4.1) as per the error boundary and initial conditions Eq. (4.2). Next, the updated approximation value turns into the following form:

4.3

This gives improved accuracy. The accuracy of updated approximations is greater than that of weighted residual approximations. To solve the above error wave Eq. (4.1), we will use fourth order CFD method. Now we will discuss the discretization process of our proposed error wave equation by using the method described by [6]. From Eq. (4.1), we get:

Let and for convenience. So we get:

4.4

Again from Eq. (4.1), we get:

4.5

Following the literature [6], we have obtained

4.6

4.7

Substituting the value from Eqs. (4.4) and (4.5) in Eqs. (4.6) and (4.7) respectively, we get

4.8

4.9

Now substituting the values of Eqs. (4.8) and (4.9) in Eq. (4.1), it was obtained,

After applying central difference formula on the L.H.S. of the above equation and doing some calculation we obtain:

4.10

for ,    where , , , ,and Here Again for using the initial conditions of the above error wave equations and applying central finite difference method, it was obtained:

4.11

Using Eq. (4.11) into Eq. (4.10) at we obtain:

4.12

Thus, the aforementioned finite difference schemes provided in Eqs. (4.10) and (4.12) result in a set of equations. By applying matrix inverse method, it provides an approximate numerical solution to the second order error wave equation implicitly.

Stability and convergence analysis

From Eq. (4.10), we have obtained:

5.1

Substituting as cited in [6] at the grid point (ih, jk) where in Eq. (5.1), we get characteristic equation:

5.2

where By Routh Hurwitz criteria [6], it was known that that if all the coefficients of the first column of Routh array are positive then the scheme is stable. Here the first column of Routh array consist of . Now we will check whether our scheme is consistent or not. Here,Clearly for sufficiently small k, . Hence the condition is satisfied. So the scheme is stable.

Here the local truncation error:

When h, then Hence the scheme is consistent with

Results and discussions

Here several second order linear and nonlinear wave problems [38, 39, 40, 41, 42–43] will be addressed using our suggested residual correction approach. If v(x, t) is the exact solution and is the approximate solution then we define some terms as follows [44]:

Absolute Error (AE) =

norm = , where are the absolute errors.

Order of Convergence, and

where and depicts norm for different space step size.

Problem 1

Consider the second order nonlinear Sine-Gordon equation from [38]:

6.1

The boundary conditions are: ,

The initial conditions:

The exact solution is [38]:

[See PDF for image]

Fig. 1

Exact solution of problem 1 over the domain [0,1] while time varies

The Fig. 1 represents a traveling wave, more specifically, the superposition of two sinusoidal waves of equal amplitude and frequency.

Table 1. Absolute errors of Problem 1 at

	Absolute error
x	Galerkin	CFD	Correction	Ref. [38]
0	0	0	0	0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1	0	0	0	0

The Table 1 shows the AE with space step size , time step size , time . The Table 1 shows that the maximum AEs obtained by the Galerkin weighted residual method where Bernstein polynomial of degree 3 are used as basis functions is and the fourth order compact finite difference approach yielded a maximum absolute error of . The maximum absolute errors by our proposed residual correction schemes is . The maximum absolute errors in reference [9] is . The Table 1 shows that the residual correction scheme gives us more precision for different values of x, and we can see that the residual correction scheme has performed better compared to the methods described in [9].

Problem 2

Let’s consider nonlinear Klein–Gordon equation from [31]:

6.2

with boundary conditions: , and initial conditions: The exact solution is [31]:

[See PDF for image]

Fig. 2

Exact solution of problem 2 over the domain while time varies

The following Fig. 2 represents a spatially linear but temporally oscillating wave. The solution varies linearly in space (since ) and cosinusoidally in time (since ). This describes an oscillating profile that maintains its shape (a straight line) but whose amplitude varies with time.

And the following Table 2 shows the absolute errors with space step size , time step size , time and then correcting the residuals using , , . The Table 2 shows that the maximum absolute error obtained by the Galerkin weighted residual method where Bernstein polynomial of degree 3 are used as basis functions is and the maximum absolute errors by fourth order compact finite difference method is , while after residual corrections it becomes .

Table 2. Absolute errors of Problem 2 at

x	Galerkin	CFD	Correction
− 1	0	0	0
− 0.8
− 0.6
− 0.4
− 0.2
0
0.2
0.4
0.6
0.8
1	0	0	0

Table 3. Maximum absolute errors of Problem 2 for different values of t

	Maximum AE
t	Correction	Ref. [31]
1
3
5
7

The above Table 3 shows the maximum absolute errors for different values of time t and the results are compared with reference [31]. Here we have taken the time step size over the interval [0, 1]. The Table 3 demonstrates that our correction results closely align with the reference results. We have obtained values at , 3, 5, 7 and in all cases, our results provide a good approximation. The result are also compared to the reference results [31].

Problem 3

Let us look at another nonlinear Klein-Gordon equation with quadratic nonlinearity from [31]:

6.3

with boundary conditions: , and initial conditions: The exact solution is [31]:

[See PDF for image]

Fig. 3

Exact solution of Problem 3 over the domain [0,1] while time varies

The above Fig. 3 describes a smooth, nonlinear growth in both space and time, reflecting the response of a medium under nonlinear forcing. The solution starts from rest—both the initial displacement and velocity are zero, but over time, the wave builds up due to the complex forcing term on the right-hand side.

Table 4. Absolute errors of problem 3 at

	Absolute error
x	Galerkin	CFD	Correction
0	0	0	0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1	0	0	0

The Table 4 shows the absolute errors with space step size , time step size , time and then correcting the residuals using , , . The maximum absolute errors obtained by the Galerkin weighted residual method where Bernstein polynomial of degree 3 are used as basis functions is and the maximum absolute errors by fourth order compact finite difference method is , while after residual corrections it becomes . The following Table 5 displays the maximum absolute errors for various values of time t and the results are compared with reference [31]. Here we have taken the time step size over the interval [0, 1]. The Table 5 demonstrates that our correction results provide much better approximation compared to the reference results. We have obtained values at , 2, 3, 4 and in all cases, our results provide a better approximation compared to the reference results [31] (Figs. 4, 5, 6, 7, 8 and 9; Tables 6, 7).

Table 5. Maximum absolute errors of Problem 3 for different values of t

	Maximum AE
t	Correction	Ref. [31]
1
2
3
4

[See PDF for image]

Fig. 4

Exact and estimated function in t = 0.05 s, with k = 0.0022, h = 0.1, for Problem 1

[See PDF for image]

Fig. 5

Exact and estimated function in t=1 s, with k = 0.0238, h = 0.1, for Problem 2

[See PDF for image]

Fig. 6

Exact and estimated function in t = 1 s, with k = 0.0238, h = 0.1, for Problem 3

[See PDF for image]

Fig. 7

Analytical and estimated error function at s, with , , for Problem 1

[See PDF for image]

Fig. 8

Analytical and estimated error function at s, with , , for Problem 2.

[See PDF for image]

Fig. 9

Analytical and estimated error function at s, with , over time for Problem 3

Table 6. Error analysis of Problem 1

	Absolute error	Convergence order
		–
		1.79
		1.60
		1.46

Table 7. Error analysis of Problem 3

	Absolute error	Convergence order
		–
		0.84
		0.93
		1.01

Thus based on our above works, we can say that the residual correction method has a high degree precision of approximate results in comparison with other methods found in the literature.

Conclusion

In this research, numerical solutions have been obtained for second-order nonlinear wave equations, followed by residual corrections. First the formulation of Galerkin weighted residual technique have been discussed where Bernstein polynomials of degree 3 has been used as basis function. Then, the formulation of fourth order compact finite difference method has been discussed. Later, we have discussed our proposed residual correction scheme for a variety of scientific and engineering applications where fourth order compact finite difference method are used to solve error wave equation. We have also discussed the stability and convergence analysis of our proposed scheme. Then we have solved several well known nonlinear Sine-Gordon equation and Klein–Gordon equation along with quadratic non-linearity and discovered, after residual correction, that the approximations of the actual solutions are pretty close. We have also compared our results to previously published results available in the literature and found that our approximations are more accurate. By using residual correction, we have obtained better approximations. All approximate results and absolute errors have been represented in tabular form. So we may conclude that our proposed scheme are highly accurate.

Acknowledgements

The authors acknowledged the University Grants Commission of Bangladesh, and the University of Dhaka, Bangladesh for supplementary support of this research.

Author contributions

MSI: Conceptualization; data curation; formal analysis; methodology; software; supervision; writing—original draft, review and editing. MH: Data curation; formal analysis; software; methodology; validation; writing—original draft. MK: Conceptualization; investigation; resources; validation; software; writing—review and editing.

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors declare no Conflict of interest.

Ethical approval and accordance

Not applicable.

Consent to publish

Not applicable.

Consent to participate

Not applicable.

Publisher's Note

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