High accuracy numerical methods for the Gardner–Ostrovsky equation

The Gardner–Ostrovsky equation, also known as the extended rotation-modified Korteweg–de Vries (KdV) equation, describes weakly nonlinear internal oceanic waves under the influence of Earth’ rotation. High accuracy numerical methods are needed to follow with precision the long time evolution of the solutions of this equation, with the additional difficulty that the numerical methods have to conserve very accurately several invariants of the solutions, including the mass and the energy of the waves. Finite-difference methods traditionally used for the solution of the KdV equation fails to preserve accurately these invariants for large times. In this paper we show that this difficulty can be overcome by using a high accuracy finite-difference (HAFD) numerical method. We present a strong-stability-preserving finite-difference scheme and compare its performance, using some relevant examples, with those of two recently published numerical methods for solving this kind of equation: a simpler second order finite-difference method and a pseudospectral numerical scheme that enforces the conservation of energy. The numerical comparison shows that the three methods have similar accuracy for short times, but the simpler finite-difference scheme is less accurate in preserving the three invariants, affecting the numerical accuracy of the solution as time goes on. On the contrary, the HAFD method presented here preserves the invariants with even better accuracy than the pseudospectral scheme, but with a much lower computational cost. In addition, the numerical implementation of the HAFD method is as easy as that of the simpler finite-difference method, being both much simpler than the intricate energy conservation pseudospectral scheme. These advantages makes the HAFD method presented here very appropriate for solving numerically this type of equations, particularly for studying long time wave propagation.