A Septic Hermite Collocation Method for Regularized Long Wave Equation

A Septic Hermite Collocation Method for Regularized Long Wave Equation Murat ARI1 1Department of Mathematics, Karamanoglu Mehmetbey University, Karaman, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0002-4039-5970 DOI : 10.5281/zenodo.18025173 Abstract In this study, a septic Hermite spline collocation method based on the classical Crank–Nicolson approximation is developed for the numerical solution of the regularized long-wave (RLW) equation. To the best of our knowledge, this represents the first application of this approach to the RLW equation. The accuracy and efficiency of the proposed scheme are thoroughly evaluated using the L2 and L∞ error norms. Numerical experiments confirm that the method is simple to implement and provides highly accurate results for the RLW equation. Keywords: Septic Hermite spline, Collocation method, RLW equation 1 Introduction In this work, we examine the following form of the regularized long wave (RLW) equation: Ut+Ux+εUUx−µUxxt = 0 (1) together with boundary conditions U(a, t) = α, U(b, t)=β, a ≤x≤b, t > 0 and the initial condition U(x, 0) = f(x). The regularized long wave (RLW) equation was originally introduced by Peregrine to represent nonlinear dispersive wave motion. It has been demonstrated that the RLW equation is capable of describing a wide range of physical processes, including nonlinear transverse waves in shallow water, ion–acoustic waves in plasma, magnetohydrodynamic waves in plasma, longitudinal dispersive waves in elastic rods, and pressure oscillations in liquid–gas bubble systems. Furthermore, the RLW equation has frequently served as a benchmark problem for numerical schemes, as it admits analytical solutions for certain sets of boundary and initial conditions [4,5,7,8,12,13,23]. The objective of the present work is to use Hermite septic spline collocation method to solve the regularized long-wave equation (RLW) numerically. This method used extensively for solving M28-1 KOSC-2025 Proceedings partial differential equations see [ 1 , 2 , 3 , 9 , 10 , 11 , 15 , 16 , 18 , 19 , 20 , 22 ] and references therein. As far as we know, this method has not been applied to RLW equation. This paper is organized as follows. First, we present the septic Hermite spline collocation method. Then, we demonstrate the application of the method to the equation. After that, numerical results and graphics are presented for some exercises. Finally, we present our conclusions in the last section. 2 The Septic Hermite Spline Collocation Method In the septic Hermite collocation method, the computational domain [ a, b ]is partitioned as π : a = x1< x2<· · · < xN+1 = b , yielding a uniform mesh with step size h = xj+1 −xj , defined by the knots xjfor j= 1,2, . . . , N + 1. The approximate solution u(x, t)is then represented in terms of septic Hermite basis functions as u(x, t)≈uN(x, t) = N+4 X j=1 aj+6k−6(t)Hji,(2) where aj ( t )are time dependent coefficients yet to be determined, k denotes the number of elements, and i = 1 , 2 , 3 , 4 , 5 , 6corresponds to the local basis functions. Within each subinterval [ xj, xj+1 ], Legendre and Chebyshev collocation points of order 6, denoted respectively by ξL i and ξC i, are employed as follows: ξL 1= 0.033765242898424, ξL 2= 0.169395306766868, ξL 3= 0.380690406958402, ξL 4= 0.619309593041598, ξL 5= 0.830604693233132, ξL 6= 0.966234757101576, and ξC 1= 0.017037086855466, ξC 2= 0.146446609406726, ξC 3= 0.37059047744874, ξC 4= 0.62940952255126, ξC 5= 0.853553390593274, ξC 6= 0.982962913144534. This formulation allows for high-order accuracy in the numerical approximation, as the septic Hermite basis ensures smoothness up to higher derivatives, while the chosen collocation nodes enhance the spectral-like precision within each finite element. The septic Hermite basis functions Hj ( x ), which satisfy the required interpolation and smoothness properties, are formulated as follows [1,2,3,20]: H6j+1(x) = 35(x−xj−1)4 h4−84(x−xj−1)5 h5+ 70(x−xj−1)6 h6−20(x−xj−1)7 h7, xj−1≤x≤xj, H6j−5(x) = 35(xj+1−x)4 h4−84(xj+1−x)5 h5+ 70(xj+1−x)6 h6−20(xj+1−x)7 h7, xj≤x≤xj+1, H6j−4(x)=−15(x−xj−1)4 h3+ 39(x−xj−1)5 h4−34(x−xj−1)6 h5+ 10(x−xj−1)7 h6, xj−1≤x≤xj, H6j+2(x)=−15(xj+1−x)4 h3+ 39(xj+1−x)5 h4−34(xj+1−x)6 h5+ 10(xj+1−x)7 h6, xj≤x≤xj+1, H6j−3(x) = 5 2 (x−xj−1)4 h2−7(x−xj−1)5 h3+13 2 (x−xj−1)6 h4−2(x−xj−1)7 h5, xj−1≤x≤xj, H6j(x) = 5 2 (xj+1−x)4 h2−7(xj+1−x)5 h3+13 2 (xj+1−x)6 h4−2(xj+1−x)7 h5, xj≤x≤xj+1, H6j−2(x)=−1 6 (x−xj−1)4 h+1 2 (x−xj−1)5 h2−1 2 (x−xj−1)6 h3+1 6 (x−xj−1)7 h4, xj−1≤x≤xj, H6j−1(x)=−1 6 (xj+1−x)4 h+1 2 (xj+1−x)5 h2−1 2 (xj+1−x)6 h3+1 6 (xj+1−x)7 h4, xj≤x≤xj+1. (3) M28-2 2nd Kocaeli Science Congress, November 19-21, 2025 To facilitate numerical computation, the system is transformed into a local coordinate framework. For each element, a local variable ξis introduced as ξ=x−xk h,(4) which maps the global interval [ xk, xk+1 ]to a fixed reference domain [0 , 1]. This normalization simplifies the evaluation of basis functions and their derivatives across all elements. Consequently, the transformation x=hξ +xkyields the following normalized forms of the basis functions: H1(ξ)=1−35ξ4+ 84ξ5−70ξ6+ 20ξ7, H2(ξ)=h(ξ−20ξ4+ 45ξ5−36ξ6+ 10ξ7), H3(ξ)=h2(0.5ξ2−5ξ4+ 10ξ5−15 2ξ6+ 2ξ7), H4(ξ)=h3(1 6ξ3−2 3ξ4+ξ5−2 3ξ6+1 6ξ7), H5(ξ)=h3(−1 6ξ4+1 2ξ5−1 2ξ6+1 6ξ7), H6(ξ)=h2(5 2ξ4−7ξ5+13 2ξ6−2ξ7), H7(ξ) = 35ξ4−84ξ5+ 70ξ6−20ξ7, H8(ξ)=h(−15ξ4+ 39ξ5−34ξ6+ 10ξ7). (5) The firstand second-order derivatives of these basis functions, denoted by Ai ( ξ )and Bi ( ξ ) respectively, are given by A1(ξ)=−140ξ3+ 420ξ4−420ξ5+ 140ξ6, A2(ξ)=h(1 −80ξ3+ 225ξ4−216ξ5+ 70ξ6), A3(ξ)=h2(ξ−20ξ3+ 50ξ4−45ξ5+ 14ξ6), A4(ξ)=h3(1 2ξ2−8 3ξ3+ 5ξ4−4ξ5+7 6ξ6), A5(ξ)=h3(−2 3ξ3+5 2ξ4−3ξ5+7 6ξ6), A6(ξ)=h2(10ξ3−35ξ4+ 39ξ5−14ξ6), A7(ξ) = 140ξ3−420ξ4+ 420ξ5−140ξ6, A8(ξ)=h(−60ξ3+ 195ξ4−204ξ5+ 70ξ6), B1(ξ)=−420ξ2+ 1680ξ3−2100ξ4+ 840ξ5, B2(ξ)=h(−240ξ2+ 900ξ3−1080ξ4+ 420ξ5), B3(ξ)=h2(1 −60ξ2+ 200ξ3−225ξ4+ 84ξ5), B4(ξ)=h3(ξ−8ξ2+ 20ξ3−20ξ4+ 7ξ5), B5(ξ)=h3(−2ξ2+ 10ξ3−15ξ4+ 7ξ5), B6(ξ)=h2(30ξ2−140ξ3+ 195ξ4−84ξ5), B7(ξ) = 420ξ2−1680ξ3+ 2100ξ4−840ξ5, B8(ξ)=h(−180ξ2+ 780ξ3−1020ξ4+ 420ξ5). (6) Finally, the trial functions and their first and second derivatives at the collocation points in terms of the local variable ξare expressed as Ui=a6k−5H1(ξi)+a6k−4H2(ξi)+a6k−3H3(ξi)+a6k−2H4(ξi)+ a6k−1H5(ξi)+a6kH6(ξi)+a6k+1H7(ξi)+a6k+2H8(ξi), hU′ i=a6k−5A1(ξi)+a6k−4A2(ξi)+a6k−3A3(ξi)+a6k−2A4(ξi)+ a6k−1A5(ξi)+a6kA6(ξi)+a6k+1A7(ξi)+a6k+2A8(ξi), h2U′′ i=a6k−5B1(ξi)+a6k−4B2(ξi)+a6k−3B3(ξi)+a6k−2B4(ξi)+ a6k−1B5(ξi)+a6kB6(ξi)+a6k+1B7(ξi)+a6k+2B8(ξi),                        i= 1(1)6.(7) This representation not only provides a smooth and continuous approximation but also allows for straightforward computation of higher-order derivatives, which is particularly advantageous in the numerical treatment of high-order partial differential equations. 3 IMPLEMENTION OF THE METHOD Using the notation Un = (x, tn) , where tn = tn−1 + ∆ t , and ∆ t is the time step, the RLW equation is discretized using Crank-Nicolson temporal scheme 2nd Kocaeli Science Congress, November 19-21, 2025 M28-3 KOSC-2025 Proceedings Un+1 −Un ∆t+Un+1 x+Un x 2+ε(UUx)n+1 + (UUx)n 2−µUn+1 xx −Un xx ∆t= 0 (8) Equation (8) can be rewritten as Un+1 −Un+∆t 2Un+1 x+Un x+ε∆t 2(UUx)n+1 + (UUx)n−µUn+1 xx −Un xx= 0 (9) The nonlinear term (UUx)n+1 in Equation (9) may be linearized by using the following term [18], (UUx)n+1 =Un+1Un x+UnUn+1 x−UnUn x So Equation (9) is discretized in time as Un+1 −µUn+1 xx +∆t 2Un+1 x+ε∆t 2Un xUn+1 +UnUn+1 x=Un−µUn xx −∆t 2Un x(10) Now substituting um values and their derivatives in (7) to the (10) and rearranging the equation yields the following matrix system of the equations: (1 + ε∆t 2Un x)H1i+ (∆t 2+ε∆t 2Un)A1i−µB1ian+1 6k−5+(1 + ε∆t 2Un x)H2i+ (∆t 2+ε∆t 2Un)A2i −µB2i)an+1 6k−4+(1 + ε∆t 2Un x)H3i+ (∆t 2+ε∆t 2Un)A3i−µB3ian+1 6k−3+(1 + ε∆t 2Un x)H4i +(∆t 2+ε∆t 2Un)A4i−µB4ian+1 6k−2+(1 + ε∆t 2Un x)H5i+ (∆t 2+ε∆t 2Un)A5i−µB5ian+1 6k−1 +(1 + ε∆t 2Un x)H6i+ (∆t 2+ε∆t 2Un)A6i−µB6ian+1 6k+(1 + ε∆t 2Un x)H7i+ (∆t 2+ε∆t 2Un)A7i −µB7i)an+1 6k+1 +(1 + ε∆t 2Un x)H8i+ (∆t 2+ε∆t 2Un)A8i−µB8ian+1 6k+2 =H1i+∆t 2A1i−µB1ian 6k−5 +H2i+∆t 2A2i−µB2ian 6k−4+H3i+∆t 2A3i−µB3ian 6k−3+H4i+∆t 2A4i−µB4ian 6k−2 +H5i+∆t 2A5i−µB5ian 6k−1+H6i+∆t 2A6i−µB6ian 6k+H7i+∆t 2A7i−µB7ian 6k+1 +H8i+∆t 2A8i−µB8ian 6k+2 i= 1,2,3,4,5,6. (11) The recursive system (11) consists of 6 N equations and 6 N + 2 unknown parameters. By applying the boundary conditions, we are able to eliminate the two parameters. This reduction results in a solvable 6N×6Nmatrix system, which is given by: Lxn+1 =Rxn where L, R coefficient matrices and, xn = (an 1,an 2,...,an 6N)T are time dependent unknown vectors to be determined. Each septic Hermite basis function in this case covers at most four subintervals, which leads to a band matrix with bandwidth six see Figure 3.1. In order to start the iterative M28-4 2nd Kocaeli Science Congress, November 19-21, 2025 process, the initial vector a 0 is needed. This vector is calculated by the initial condition presented by the governing equation.                     ×××××××··········· ×××××××··········· ×××××××··········· ×××××××··········· ×××××××··········· ×××××××··········· · · · · · × × × × × × × × · · · · · · · · · · × × × × × × × × · · · · · · · · · · × × × × × × × × · · · · · · · · · · × × × × × × × × · · · · · · · · · · × × × × × × × × · · · · · · · · · · × × × × × × × × · · · · · ·················· ·················· ·················· ···········××××××× ···········××××××× ···········××××××× ···········××××××× ···········××××××× ···········×××××××                     Figure 3.1: Patterns of nonzero elements of banded matrixes L and R. 4 Numerical Examples In this section, we use two test problems with known exact results to analyse how accurate and reliable the proposed method is numerically. The accuracy of the method is also evaluated using two basic error measures square error L2and maximum error L∞norms: L2=v u u th N X j=1 uexact j−unum. j 2, L∞= max 1≤j≤Nuexact j−unum. j. The RLW equation has three conservation quantities corresponding to mass, momentum and energy C1=Zb a udx ≃h N X j=1 un j C2=Zb ahu2+µ2(ux)2idx ≃h N X j=1 un j2+hµ(ux)n ji2 C3=Zb ahεu3+ 3u2idx ≃h N X j=1 εun j3+ 3un j2 respectively, and these will be monitored to check the conserved properties of the numerical algorithm. Example 4.1. Single solitary wave The exact solution to the RLW equation (1) is given by the expression: U(x, t) = 3dsech2(k[x−x0−vt]) ,(12) This formula describes a single solitary wave possessing an amplitude of 3 d , a velocity of v = 1+ εd , and k = 1 2 ( εd/µν ) 1/2 . The wave’s propagation is modeled over the spatial domain − 40 ≤x≤ 60 and the time interval 0 ≤t≤ 20, utilizing the parameters ε = µ = 1 and x0 = 0. The numerical 2nd Kocaeli Science Congress, November 19-21, 2025 M28-5 KOSC-2025 Proceedings simulation is initiated with the following initial condition: U(x, 0) = 3dsech2(k[x−x0]) .(13) Table 1summarizes the findings derived from applying the septic Hermite collocation method, which employs both Legendre and Chebyshev functions. The resulting numerical data exhibits exceptional accuracy. Furthermore, Figure 4.1 provides a visual representation of the trajectory of the single solitary wave solution for the RLW equation, specifically for the parameter values d= 0.01 and d= 0.03. Method h∆t L2Error L∞Error C1C2C3 d= 0.1 Analytic 3.9799497 0.81046249 2.5790074 SHCM-L 1 0.1 5.10789 ×10−47.80226 ×10−53.9800198 0.81046313 2.5790074 SHCM-C 0.1 5.14400 ×10−47.80301 ×10−53.9800552 0.81046614 2.5790074 MQ[6] 1 0.1 2.08318 ×10−47.8619 ×10−53.979960 0.8104726 2.579041 TPS[6] 0.1 2.96994 ×10−48.5119 ×10−53.979823 0.8104159 2.578853 IQ[6] 0.1 2.18123 ×10−47.6040 ×10−53.979582 0.8104589 2.578995 G[6] 0.1 2.06232 ×10−47.5355 ×10−53.979846 0.8104508 2.578968 d= 0.03 Analytic 2.1094075 0.12730718 0.3888059 SHCM-L 1 3.20534 ×10−34.43070 ×10−42.1122781 0.13138396 0.3888119 SHCM-C 4.54963 ×10−38.56998 ×10−52.1142600 0.14479677 0.3888175 MQ[6] 1 0.1 4.0259 ×10−51.1922 ×10−52.107167 0.1273011 0.388804 TPS[6] 0.1 6.45907 ×10−44.30236 ×10−42.105547 0.1273215 0.388865 IQ[6] 0.1 7.45694 ×10−44.31517 ×10−42.104109 0.1273011 0.388802 G[6] 0.1 2.87437 ×10−48.3060 ×10−52.106074 0.1272980 0.388795 Table 1: Comparison of error norms and constants for various methods for ∆ t = 0 . 1, and t = 20. Figure 4.1: The plots of Septic Hermite Spline solutions of RLW with a,d= 0.1,b,d= 0.03. Example 4.2. Interaction of two waves For the second test problem concerning the RLW equation, the interaction of two solitary waves is selected. The initial condition for the interaction of two positive solitary waves is defined as the sum of two well-separated solitary waves of differing amplitudes: M28-6 2nd Kocaeli Science Congress, November 19-21, 2025 U(x, 0) = U1+U2 Uj= 3Ajsech2(kj(x−˜xj)), Aj=4k2 j 1−4k2 j , j = 1,2(14) The problem is constrained by the boundary conditions U (0 , t ) = U (120 , t ) = 0. The chosen parameters are k1 = 0 . 4, ˜x1 = 15, k2 = 0 . 3, and ˜x2 = 35 . 1. These values result in two solitary waves with approximate magnitudes of 5 . 3338 and 1 . 6860, whose peaks are initially positioned at x = 15 and x = 35 . 1, respectively. Consequently, the larger wave is initially situated to the left of the smaller wave. The simulation is performed over the spatial interval 0 ≤x≤ 120 up to time t= 30. The numerical solutions illustrating the wave interaction are presented in Figure 4.2, obtained using the aforementioned Hermite spline scheme. Both solitary waves travel to the right, with their velocities determined by their respective amplitudes. The interaction between the two waves took place at approximately t = 15. Significantly, both waves fully regained their original shapes after the interaction, which is a characteristic property of solitons. Table 2: Invariants of two soliton problem at different times. Method t C1C2C3 CHCM-L 5 37.9172508543 120.5225776494 744.0569871365 CHCM-C 5 37.9172357037 120.5299525086 744.0570118913 CHCM-L 10 37.9179233337 120.4649085281 743.4582503203 CHCM-C 10 37.9179942258 120.4689959629 743.4582227092 CHCM-L 15 37.9185694486 120.3186589219 741.9047193489 CHCM-C 15 37.9188775106 120.3208055064 741.9047161537 CHCM-L 20 37.9192042267 120.5088058791 743.9314947268 CHCM-C 20 37.9198739896 120.5098735376 743.9315382605 CHCM-L 25 37.9198248648 120.5222139975 744.0713532892 CHCM-C 25 37.9209626300 120.5227186165 744.0713772404 CHCM-L 30 37.9203892813 120.5226402231 744.0756579395 CHCM-C 30 37.9220721085 120.5228691038 744.0756327904 2nd Kocaeli Science Congress, November 19-21, 2025 M28-7 KOSC-2025 Proceedings Figure 4.2: Collision of two solitons. 5 CONCLUSION This study investigates numerical solutions for the nonlinear RLW equation using the hermite septic spline method. The method provide a simple way to approximate solutions for similiar equations. 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Karakaş, A powerful robust cubic Hermite collocation method for the numerical calculations and simulations of the equal width wave equation, arXiv preprint, (2023) arXiv:2309.02439. https://arxiv.org/abs/2309.02439 2nd Kocaeli Science Congress, November 19-21, 2025 M28-9