Convergence of linearized sequence tasks to the nonlinear sediment transport task solution

Present paper has been devoted to convergence investigation of solution of $\mathrm{2D}$ linearized bottom sediment transportation task sequence to the solution of appropriate nonlinear task in the Hilbert space $L_1$, when the value of time step tends to $0$. These models are taking into account: wave influence, bottom relief, density and porosity sediment deposits and tangential tension near the bottom for coastal zone. In previous author investigations the existence and uniqueness of $\mathrm{2D}$ linearized bottom sediment transportation task sequence have been proved and the estimation in Hilbert space $L_1$ has been done for appropriate boundary value problem in dependence of integral expressions for right side function, boundary and initial conditions. In previous author papers have been obtained results for constructed finite difference scheme convergence and stability, approximated the sequence of linearized bottom sediment transportation task. Earlier numerical results for model and real tasks have been presented for coastal zone. In given work have been presented elaboration results for convergence of linearized boundary value problem sequence solution to the solution of appropriate nonlinear task, when time step approaches to $0$. The fact of convergence of linear difference scheme solution to the solution of appropriate nonlinear bottom sediment transportation task is proved, when time and space steps tend to zero in according of results of given and previous papers.