Coupling of the Crank–Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow

This paper proposes an efficient localized meshless technique for approximating the viscoelastic wave model. This model is a significant methodology to explain wave propagation in solids modeled with a wide collection of viscoelastic laws. In the first method, a difference scheme with the second-order accuracy is implemented to obtain a semi-discrete scheme. Then, a localized radial basis function partition of unity scheme is adopted to get a full-discrete scheme. This localization technique consists of decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain. A well-conditioned resulting linear system and a low computational burden are the main merits of this technique compared to global collocation methods. Further, the stability and convergence analysis of the temporal discretization scheme are deduced using discrete energy method. Numerical results are shown to validate the accuracy and effectiveness of the proposed method.