Solution of the generalized regularized long-wave equation with optimal spline collocation technique and implicit Crank–Nicolson as well as explicit SSP-RK43 scheme

Abstract

In this work, a generalized regularized long-wave equation is solved using the optimal cubic B-spline collocation method for space discretization. Two different approaches are followed for the time-domain discretization. One is the implicit Crank–Nicolson scheme and the other is strong explicit stability preserving Runge–Kutta method of four stages and third-order. Also, the stability analysis of the techniques is carried out and it is shown that the implicit technique is unconditional stable, whereas the explicit technique is conditionally stable. These methods are applied to three problems involving a single solitary wave, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. These equations possess three invariants of motion that are mass, momentum, and energy. The value of these invariants is calculated, which is found to remain preserved for a long time. To demonstrate the robustness of both the techniques, the $L_2$ and $L_{\mathrm{\infty }}$ error norms are calculated.

Keywords:

• GRLW equation
• cubic B-splines
• optimal collocation method
• Crank–Nicolson
• SSP-RK43 scheme

AMS SUBJECT CLASSIFICATION CODE:

• 34B05
• 65L10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Ms. Shallu is thankful to CSIR New Delhi for providing financial assistance in the form of SRF with File No. 09/797(0016)/2018-EMR-I.