Finite difference/spectral approximations to a water wave model with a nonlocal viscous term

The numerical simulation of a water wave model with a nonlocal viscous dispersive term is considered in this article. We construct two linearized finite difference/spectral schemes for numerically solving the considered water model. A particular attention is paid to the treatment of the nonlocal dispersive term and the nonlinear convection term. The proposed methods employ a known (2 - alpha)-order scheme for the alpha-order fractional derivative and a mixed linearization for the nonlinear term. A detailed analysis shows that the proposed schemes are unconditionally stable. Some error estimates are provided to predict that the method using the linearized Euler plus (2 - alpha)-order scheme in time and the spectral approximation in space is convergent with order of O(Delta t + N1-m), where Delta t, N and m are, respectively the time step size, polynomial degree, and regularity in the space variable of the exact solution. Moreover, we prove that the second order backward differentiation plus (2 - alpha)-order scheme converges with order 3/2 in time. A series of numerical examples is presented to confirm the theoretical prediction. Finally the proposed methods are used to investigate the asymptotic decay rate of the solutions of the water wave equation, as well as the impact of different terms on this decay rate. (C) 2014 Elsevier Inc. All rights reserved.