High Order Finite Difference/Spectral Methods to a Water Wave Model with Nonlocal Viscosity (Journal of Computational Mathematics, 2020)
In this paper, efficient numerical schemes are proposed for solving the water wave modelwith nonlocal viscous term that describe the propagation of surface water wave. By usingthe Caputo fractional derivative definition to approximate the nonlocal fractional operator,finite difference method in time and spectral method in space are constructed for theconsidered model. The proposed method employs known 5/2 order scheme for fractionalderivative and a mixed linearization for the nonlinear term. The analysis shows thatthe proposed numerical scheme is unconditionally stable and error estimates are providedto predict that the second order backward differentiation plus 5/2 order scheme convergeswith order 2 in time, and spectral accuracy in space. Several numerical results are providedto verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate ofsolutions is investigated.
Keywords: Water waves; Nonlocal viscosity; Finite difference; Spectral method; Convergence order; Decay rate.