Benchmark travelling waves and bidirectional adaptive DA–PMA computation for the two-dimensional regularized long-wave equation

H. S. Alayachi; A. R. Alharbi; H. S. Alayachi; A. R. Alharbi

doi:10.3934/math.2026708

AIMS Mathematics

2026, Volume 11, Issue 6: 17293-17319. doi: 10.3934/math.2026708

Previous Article Next Article

Research article Special Issues

Benchmark travelling waves and bidirectional adaptive DA–PMA computation for the two-dimensional regularized long-wave equation

H. S. Alayachi ,
A. R. Alharbi ^,

Department of Mathematics, College of Science, Taibah University, Medina 42353, Saudi Arabia

Received: 08 April 2026 Revised: 21 May 2026 Accepted: 27 May 2026 Published: 15 June 2026
MSC : 35C07, 35Q35, 35Q53, 65M06, 65M12, 65M50

This paper presents an adaptive computational framework for the two-dimensional regularized long-wave (RLW) equation. The main focus was on the numerical treatment of the model on moving nonuniform meshes. The RLW equation was first rewritten in a mixed differential-algebraic (DA) form by introducing an auxiliary variable. This reformulation avoids the direct discretization of the mixed space–time derivatives and leads to a more convenient adaptive implementation. The spatial approximation was constructed on bidirectionally adaptive tensor-product grids using five-point Fornberg finite-difference operators. Mesh redistribution was performed through a parabolic Monge–Ampère (PMA) rezoning strategy combined with monitor functions and shape-preserving interpolation between successive meshes. Exact travelling-wave solutions were also derived through an oblique wave reduction. These solutions were used mainly as benchmark data for the numerical computations. In particular, the solitary-wave solution provides compatible initial conditions, boundary data, auxiliary-field values, and reference profiles for error measurement. The stability discussion was carried out under admissible mesh evolution. On each frozen mesh, the spatial approximation was formally second-order consistent. The convergence behavior of the full adaptive algorithm was assessed numerically because mesh redistribution and inter-mesh interpolation introduce additional error effects. Numerical experiments were presented for single-wave and two-pulse configurations. The results show that the adaptive method preserves the wave structure with good accuracy and produces smaller errors than the corresponding fixed-mesh computation. The adaptive meshes remained concentrated near the dominant wave regions while maintaining global mesh regularity throughout the simulation.
Citation: H. S. Alayachi, A. R. Alharbi. Benchmark travelling waves and bidirectional adaptive DA–PMA computation for the two-dimensional regularized long-wave equation[J]. AIMS Mathematics, 2026, 11(6): 17293-17319. doi: 10.3934/math.2026708

Related Papers:

Abstract

This paper presents an adaptive computational framework for the two-dimensional regularized long-wave (RLW) equation. The main focus was on the numerical treatment of the model on moving nonuniform meshes. The RLW equation was first rewritten in a mixed differential-algebraic (DA) form by introducing an auxiliary variable. This reformulation avoids the direct discretization of the mixed space–time derivatives and leads to a more convenient adaptive implementation. The spatial approximation was constructed on bidirectionally adaptive tensor-product grids using five-point Fornberg finite-difference operators. Mesh redistribution was performed through a parabolic Monge–Ampère (PMA) rezoning strategy combined with monitor functions and shape-preserving interpolation between successive meshes. Exact travelling-wave solutions were also derived through an oblique wave reduction. These solutions were used mainly as benchmark data for the numerical computations. In particular, the solitary-wave solution provides compatible initial conditions, boundary data, auxiliary-field values, and reference profiles for error measurement. The stability discussion was carried out under admissible mesh evolution. On each frozen mesh, the spatial approximation was formally second-order consistent. The convergence behavior of the full adaptive algorithm was assessed numerically because mesh redistribution and inter-mesh interpolation introduce additional error effects. Numerical experiments were presented for single-wave and two-pulse configurations. The results show that the adaptive method preserves the wave structure with good accuracy and produces smaller errors than the corresponding fixed-mesh computation. The adaptive meshes remained concentrated near the dominant wave regions while maintaining global mesh regularity throughout the simulation.

References

[1]	D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321–330. https://doi.org/10.1017/S0022112066001678 doi: 10.1017/S0022112066001678
[2]	T. B. Benjamin, J. L. Bona, J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. A Math. Phys. Eng. Sci., 272 (1972), 47–78. https://doi.org/10.1098/rsta.1972.0032 doi: 10.1098/rsta.1972.0032
[3]	J. A. Goldstein, B. J. Wichnoski, On the Benjamin–Bona–Mahony equation in higher dimensions, Nonlinear Anal., 4 (1980), 665–675. https://doi.org/10.1016/0362-546X(80)90067-X doi: 10.1016/0362-546X(80)90067-X
[4]	J. Avrin, J. A. Goldstein, Global existence for the Benjamin–Bona–Mahony equation in arbitrary dimensions, Nonlinear Anal., 9 (1985), 861–865. https://doi.org/10.1016/0362-546X(85)90023-9 doi: 10.1016/0362-546X(85)90023-9
[5]	P. J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Philos. Soc., 85 (1979), 143–160. https://doi.org/10.1017/S0305004100055572 doi: 10.1017/S0305004100055572
[6]	L. A. Medeiros, M. M. Miranda, Weak solutions for a nonlinear dispersive equation, J. Math. Anal. Appl., 59 (1977), 432–441. https://doi.org/10.1016/0022-247X(77)90071-3 doi: 10.1016/0022-247X(77)90071-3
[7]	J. L. Bona, W. G. Pritchard, L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. A Math. Phys. Eng. Sci., 302 (1981), 457–510. https://doi.org/10.1098/rsta.1981.0178 doi: 10.1098/rsta.1981.0178
[8]	J. L. Bona, W. R. McKinney, J. M. Restrepo, Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation, J. Nonlinear Sci., 10 (2000), 603–638. https://doi.org/10.1007/s003320010003 doi: 10.1007/s003320010003
[9]	B. Tian, W. Li, Y. T. Gao, On the two-dimensional regularized long-wave equation in fluids and plasmas, Acta Mech., 160 (2003), 235–239. https://doi.org/10.1007/s00707-002-0967-0 doi: 10.1007/s00707-002-0967-0
[10]	W. X. Ma, B. Fuchssteiner, Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. I, Int. J. Non-Linear Mech., 31 (1996), 329–338. https://doi.org/10.1016/0020-7462(95)00064-X doi: 10.1016/0020-7462(95)00064-X
[11]	V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Springer-Verlag, 1991. https://doi.org/10.1007/978-3-662-00922-2
[12]	A. R. Alharbi, Traveling-wave and numerical solutions to a Novikov–Veselov system via the modified mathematical methods, AIMS Math., 8 (2023), 1230–1250. https://doi.org/10.3934/math.2023062 doi: 10.3934/math.2023062
[13]	A. R. Alharbi, Traveling-wave and numerical solutions to nonlinear evolution equations via modern computational techniques, AIMS Math., 9 (2024), 1323–1345. https://doi.org/10.3934/math.2024065 doi: 10.3934/math.2024065
[14]	A. Ahmed, A. R. Alharbi, I. Hashim, Exact and numerical solutions of the generalized breaking soliton system: Insights into non-linear wave dynamics, AIMS Math., 10 (2025), 5124–5142. https://doi.org/10.3934/math.2025235 doi: 10.3934/math.2025235
[15]	A. Ahmed, A. R. Alharbi, H. S. Alayachi, I. Hashim, Exact and numerical approaches for solitary and periodic waves in a (2+1)-dimensional breaking soliton system with adaptive moving mesh, AIMS Math., 10 (2025), 8252–8276. https://doi.org/10.3934/math.2025380 doi: 10.3934/math.2025380
[16]	A. Ahmed, T. G. Alharbi, A. R. Alharbi, I. Hashim, Traveling wave reductions and adaptive moving mesh computations for the improved Boussinesq equation, AIMS Math., 10 (2025), 28374–28395. https://doi.org/10.3934/math.20251248 doi: 10.3934/math.20251248
[17]	I. Alraddadi, F. Alsharif, S. Malik, H. Ahmad, T. Radwan, K. K. Ahmed, Innovative soliton solutions for a (2+1)-dimensional generalized KdV equation using two effective approaches, AIMS Math., 9 (2024), 34966–34980. https://doi.org/10.3934/math.20241664 doi: 10.3934/math.20241664
[18]	F. Calogero, A. Degasperis, Spectral transform and solitons: tools to solve and investigate nonlinear evolution equations, North-Holland, 1982.
[19]	K. Spayd, M. Shearer, The Buckley–Leverett equation with dynamic capillary pressure, SIAM J. Appl. Math., 71 (2011), 1088–1108. https://doi.org/10.1137/100807016 doi: 10.1137/100807016
[20]	R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527–543. https://doi.org/10.1137/0503051 doi: 10.1137/0503051

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)