On Numerical Solution of the Gardner–Ostrovsky Equation

¹ E.T.S. Ingeniería Industrial, University of Malaga, Dr Ortiz Ramos s/n, 29071, Malaga, Spain
² Department of Mathematics and Computing, Faculty of Sciences, University of Southern Queensland, Toowoomba, Australia

^⋆ Corresponding author. E-mail: Yury.Stepanyants@usq.edu.au

Abstract

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovsky equation (u_t + cu_x + α uu_x + α₁u²u_x + βu_xxx)_x = γu which is also known as the extended rotation-modified Korteweg–de Vries (KdV) equation. This equation is used for the description of internal oceanic waves affected by Earth’ rotation. Particular versions of this equation with zero some of coefficients, α, α₁, β, or γ are also known in numerous applications. The proposed numerical scheme is a further development of the well-known finite-difference scheme earlier used for the solution of the KdV equation. The scheme is of the second order accuracy both on temporal and spatial variables. The stability analysis of the scheme is presented for infinitesimal perturbations. The conditions for the calculations with the appropriate accuracy have been found. Examples of calculations with the periodic boundary conditions are presented to illustrate the robustness of the proposed scheme.