Numerical method for the propagation of BBMB model with small and medium amplitude long waves in nonlinear dispersive media

When simulating the unidirectional propagation of plane waves, the Benjamin– Bona–Mahony–Burgers (BBMB) equation is a very important mathematical model considering the effects of dissipation mechanisms in addition to nonlinearity and dispersion mechanisms. Based on the Thin-Plate-Spline-Radial-Basis-Function collocation method, we propose a single-layer meshless numerical method within a spacetime collocation framework for the propagation of a 1D BBMB model with small and medium amplitude long waves in nonlinear dispersive media. This meshless algorithm adopts the direct solution procedure of the collocation method. It breaks through the limitations of traditional two-layer solution procedures by introducing combined Thin-Plate-Spline-Radial-Basis-Functions within the spacetime collocation framework. It is shown that nonlinear 1D BBMB equations with time-varying characteristics can be transformed into single-layer problems. Numerical examples have verified the advantages of this method in terms of computational accuracy and efficiency. Compared with traditional two-layer numerical methods, the proposed single-layer solving process can maintain or even exceed the accuracy of traditional methods.