TY - JOUR

T1 - A comparative study of bi-directional Whitham systems

AU - Dinvay, Evgueni

AU - Dutykh, Denys

AU - Kalisch, Henrik

N1 - Funding Information:
This research was supported in part by the Research Council of Norway through grants 213474/F20 and 239033/F20 .
Publisher Copyright:
© 2018 IMACS

PY - 2019/7

Y1 - 2019/7

N2 - In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties. In the current work, we review some of the existing fully dispersive systems, such as found in [1,4,9,17,22,23]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

AB - In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties. In the current work, we review some of the existing fully dispersive systems, such as found in [1,4,9,17,22,23]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

KW - Fully dispersive system

KW - Split-step scheme

KW - Surface water waves

KW - Whitham equation

UR - http://www.scopus.com/inward/record.url?scp=85054163031&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2018.09.016

DO - 10.1016/j.apnum.2018.09.016

M3 - Article

AN - SCOPUS:85054163031

SN - 0168-9274

VL - 141

SP - 248

EP - 262

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -