Numerical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach

Abstract

Communicated by: R. RODRIGUEZ Funding information PHC Utique ASEO MSC Classification: 35Q35; 35Q53In this paper, we numerically study the water wave model with a nonlocal viscous termds is the Riemann-Liouville half-order derivative in time. We propose and compare different numerical schemes based on the diffusive realizations of fractional operators.

Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

“…have respectively been considered, to model natural damping of water waves, mathematical analysis and simulations can be found also in [12,23,30] in which…”

“…was considered. The mathematical analysis is given in [23] and a numerical study is presented in [30]. • A localized damping in space corresponds to 3.2.…”

“…The mathematical analysis of the long time behavior of the solutions of the resulting models is also essential to the understanding of the underlying physics [23,42,43,44,45,48]. Of course the derivation of appropriate and robust numerical schemes is crucial to capture the dynamics and also to point out mathematical properties that are difficult to establish, [12,21,30,40]. It is to be noticed that the presence of a damping term can be seen as a stabilization technique used in control theory, see [50,59].…”

Decay of solutions to a water wave model with a nonlocal viscous term

“…Following [6], we use the Gauss-Jacobi quadrature method with N m points to approximate the generalized integral in (3.2). We note by w i the weights and by σ i the nodes of the Gauss-Jacobi quadrature method.…”

“…where G 1 is a diagonal matrix of order N m + 1. Following [6], we use a splitting method to provide numerical results. Let t > 0, for all n ≥ 0, we recall that t n = n t and…”