Numerical Analysis of the Discrete MRLW Equation for a Nonlinear System Using the Cubic B-Spline Collocation Method

: By employing the cubic B-spline functions, a collocation approach was devised in this study to address the Modi ﬁ ed Regularized Long Wave (MRLW) equation. Then, we derived the corresponding nonlinear system and easily solved it using Newton’s iterative approach. It was established that the cubic B-spline collocation technique exhibits unconditional stability. The dynamics of solitary waves, including their pairwise and triadic interactions, were meticulously investigated utilizing the proposed numerical method. Additionally, the transformation of the Maxwellian initial condition into solitary wave formations is presented. To validate the current work, three distinct scenarios were compared against the analytical solution and outcomes from alternative methods under both 2 L - and ∞ L -error norms. Primarily, the key strength of the suggested scheme lies in its capacity to yield enhanced numerical resolutions when employed to solve the MRLW equation, and these conservation laws show that the solitary waves have time and space translational symmetry in the propagation process. Finally, this paper concludes with a summary of our ﬁ ndings.


Introduction
Nonlinear partial differential equations are of great importance in elucidating various phenomena in all sorts of fields.However, exact solutions are not available for all such equations.Thus, it is necessary to employ different numerical methods to explore their possible solutions.
The form of a regularized long wave (RLW) equation is as follows: The equation, characterized by the positive parameters δ and μ , was initially put forth by Peregrine [1] to illustrate the dynamics of an undular bore, according to the symmetry and conservation law of time translation.He pioneeringly employed the finite difference method to formulate the first numerical approach for solving the RLW equation.This equation specifically encompasses events characterized by slight nonlinearity and dispersion, such as the nonlinear lateral waves occurring in shallow waters, ion-acoustic and magnetohydrodynamic disturbances within plasmas, and phonon propagations in nonlinear crystalline structures.This equation's existence and singular-Citation: Liu, X.; Zhang, L.; Sun, J. ity were identified by Refs.[2,3] in 1973.Later, Refs.[4][5][6][7][8][9][10] and other scholars contributed to the topic by exploring its numerical resolutions from the perspectives of the finite difference and finite element method.
So far, a multitude of numerical approaches, such as finite element methods and analytical solution strategies, have been proposed for addressing the RLW equations.According to references [11][12][13][14][15], the finite difference approaches and finite element methodologies like collocation methods that utilize quadratic, cubic, and up-to-date septic B-splines have been employed to resolve the RLW equation.Indeed, the RLW equation is a special case of the generalized long wave (GRLW) equation that has the following form: This equation is characterized by a positive integer p in this paper.The General- ized Regularized Long Wave (GRLW) equation has attracted little attention from researchers, with Zhang [16] exploring it through a finite difference approach to addressing a Cauchy problem and Kaya [17] employing the Adomian Decomposition Method (ADM).This work, however, delves into a distinct variant of the GRLW equation known as the Modified Regularized Long Wave (MRLW) equation.Numerical resolutions for the MRLW equation have been developed using various techniques.The finite difference method was utilized in [18], while [19,20] applied collocation methods with quintic B-splines.In a separate study, Refs.[21][22][23] employed cubic B-splines for the same purpose.The cubic B-spline Galerkin finite element method was then adopted to derive numerical solutions in [24][25][26].The authors discuss the mixed finite element method for solving a coupled wave equation in [27] and for solving a damped Boussinesq equation in [28].
In this scheme, we continue to use the cubic B-splines for the same purpose, but we focus on temporal discrete values with a half time layer, and it is found that our scheme has better accuracy and stability than that presented in [20] for three solitary waves and for Maxwellian pulse splitting in the numerical simulation, indicating that our scheme satisfies the conservation laws perfectly.
In this study, a cubic-B-spline-based collocation technique was employed to address the MRLW equation.Section 2 details the collocation approach employed for resolving this equation, while Section 3 presents an examination of the method's linear stability.To validate its precision, analytical solutions and conservation laws are utilized in the assessment process.The dynamics of solitary waves, including their interactions and the transformation of Maxwellian initial conditions into solitary waves, are investigated through numerical simulations in Section 4. Finally, conclusions are drawn in the closing section.

Governing Equation and Cubic B-Spline Collocation Method
The MRLW equation is expressed as follows: Here, the notations x and t symbolize differentiation, with the equation subject The analytical solution of this equation can be written as Here, j c denotes unforeseen time-varying entities that are to be derived through the collocation approach, considering the constraints of boundary and initial conditions.
The cubic-B spline functions are as follows.
By employing Equation ( 6), the estimation of nodal values j c , along with the first and second derivatives j c′ and j c ′ ′ at the nodal points, can be readily derived.
). 2 ( 6), ( 3 Let us write the MRLW equation in the following form: Following the method used by S.I.Zaki [21] to solve the the KdVB equation, we adapt his technique to derive a recursive formula for numerically resolving Equation ( 8), with the temporal focus set at . Here, t Δ represents the time interval, and we employ a Crank-Nicholson methodology: Here, the notations n and n + 1 denote the chronological sequence of time steps.These computations correspond to second-order precise estimations of the values occurring at the specific time instance.Then, Equation (8) becomes Introducing Equation (7) into Equation ( 12 where The system that emerges from Equation ( 13) is of a nonlinear nature.It can be effectively addressed through Newton's iterative approach, typically achieving convergence after just two iterations.

Linear Stability Analysis
In order to examine the equilibrium stability of Equation ( 13) when linearized, we employ the Von Neumann methodology.The nonlinear term x u u 2 is linearized by using u as a constant σ .Let n n ikjh j c q e = ; by incorporating this into Equation (13), we obtain After some manipulations, the following can be obtained: The amplification factor is given by So, this scheme is unconditionally stable.

Numerical Examples and Results
Evaluating numerical methods for evolutionary equations involves validating specific characteristics; these methods must faithfully reproduce the chronological evolution of solitary wave behavior.Throughout their propagation, the solution must maintain compliance with equivalent conservation principles.The accuracy in the movement of solitary waves can be objectively measured through the calculation of both the 2 L -error norm and These characteristics facilitate the assessment of numerical methods, particularly in scenarios where analytical solutions are non-existent and during the dynamics of soliton collisions.

Single Solitary Waves
In case I, we set c = 0. − at t = 20.Table 1 exhibits the findings pertaining to case II.The visual representation of the solitary wave's progression at various temporal stages is depicted in Figure 1.A graphical comparison of the discrepancies between the analytical and computational solutions at t = 20 can be found in Figure 2, where the highest absolute error value surfaces approximately at the crest of the solitary wave.Furthermore, when comparing the results obtained using the proposed method with the results from [19,20] in Table 2, we find that when the solitary wave has an amplitude of 0.54772 (c = 0.3), our method provides the same results as the cubic B-spline collation methods [20].

Interaction of Two Solitary Waves
The second segment of this numerical investigation explores the interaction dynamics of two solitary waves within the context of the MRLW equation.The initial scenario is established by considering a linear superposition of two distinct, amplitude-varied solitary waves that are spatially well-distinguished. )) where This experiment observes the interaction of dual solitary waves, differing in amplitude yet propagating co-directionally.Upon collision, they proceed to reemerge without any alteration in their initial states.The solitary wave interaction problem was solved in the interval [0, 250] for t = 0 to t = 20 with  3 depicts the invariants across various temporal stages.A graph- ical representation of the interaction among these solitary waves at distinct time levels is depicted in Figure 3.

Interaction of Three Solitary Waves
This section focuses on investigating the dynamic interaction between three distinct-amplitude MRLW solitons moving co-directionally.This examination is carried out using the MRLW equation, which incorporates initial conditions that represent a linear combination of three well-spaced solitary waves, each with a unique amplitude. )) where In our computational study, we opt for the following parameters: x = 60, h = 0.2, and t Δ = 0.025, with the domain spanning from 0 to 250.The amplitude proportions are set at a ratio of 4:2:1; it is note- Our computational study reveals that the invariants 1 I , 2 I , and 3 I for the solitary wave collisions remain constant, even with significant wave magnitudes.These observations are documented in Table 4.The intricate dynamics of these solitary waves at various temporal stages are depicted in Figure 4, extending up to t = 45.
As a train of solitary waves, it is applied to the problem for different cases: (I) μ = 0.1, (II) μ = 0.04, (III) μ = 0.015, and (IV) μ = 0.01.The simulations are conducted up to t = 15.The values of the quantities 1 I , 2 I , and 3 I are given in Table 5.However, when μ is reduced, more solitary waves formed.For case (I), only a single soliton is generated, as shown in Figure 5a, while for case (II), the Maxwellian pulse develops into a train of at least two solitary waves, as shown in Figure 5b.Similarly, Figure 5c shows that for case (III), three stable solitons are generated, and Figure 5d indicates that for case (IV), the Maxwellian initial condition has decayed into four stable solitary waves.The peaks of the well-developed wave lie on a straight line, so their velocities are linearly dependent on their amplitudes and a small oscillating tail appearing behind the last wave, as shown in Figure 5.The values of the quantities 1 I , 2 I , and 3 I for cases μ = 0.1, 0.04, 0.015, and 0.01 are given in Table 5.   vides smaller maximum changes of the three invariants than the cubic B-spline collocation method [20], which indicates that our scheme is satisfactorily conservative.The following is a comparison of the numerical results: In the simulation of the 4.1 solitary wave and the 4.2 double solitary wave, the results of the two algorithms are equivalent, but in the simulation of the 4.3 triple solitary wave and the 4.4 Maxwellian initial condition, our algorithm's result is closer to the analytical solution, and the maximum variation of the conserved quantity is smaller, indicating that our algorithm has higher accuracy and better satisfies the conservation law of the equation.

Conclusions
A collocation approach employing cubic B-splines was established for generating solitary waves in the context of the MRLW equation.It was found that this approach exhibits marginal stability and surpasses other cubic B-spline collocation methods in terms of precision [19].It was also shown in the computed examples provided in this paper that the conservation laws are substantially satisfied, where the maximum changes of three invariants in our scheme are all smaller than those in the collocation scheme [20].We propose that these algorithms can be used to obtain numerical solutions to nonlinear differential equations.Most importantly, the use of the collocation method is especially advisable for deriving the solitary waves of nonlinear differential equations containing high-power nonlinear terms.

.
For the scope of this research, periodic boundary conditions are adopted within the domain b

x
and c are arbitrary constants.Partition the interval [a,b] at points by j cubic B-splines collectively constitute a foundation of functions that span across the interval [a, b].A global approximation solution terms of the cubic B-splines and unknown time-dependent parameters as follows: for all values of k.

1 A = 2 2 A
55, a space step h = 0.2, and time step t Δ = 0.025; then, the amplitudes are in a ratio of 2:1, where .Table

Table 1 .
Invariants and errors for a single solitary wave with c = 0.3.

Table 2 .
Errors and invariants of single solitary wave.

Table 3 .
Invariants for interaction of two solitary waves.

Table 4 .
Invariants for interaction of three solitary waves.

Table 5 .
Invariants of MRLW equation using the Maxwellian initial condition.

Table 6
[20]ents a comparison between the cubic B-spline collocation approach[20]and our scheme according to the maximum changes 1 2 I , and 3 I in the above computed examples.It shows that, generally, our scheme pro-

Table 6 .
The maximum changes of invariants 1