Sufficient convergence conditions for positive solutions of linearized two-dimensional sediment transport problem

Introduction. The sediment transportation is the one of major processes that defines the magnitude and bottom surface changing rate of water basins. The most commonly used predictable researches in this field are based on mathematical models. Modeling gives possibilities to reduce and in some cases - to eliminate expensive and often dangerous experiments. Spatially one-dimensional models have been usually used to predict changes of water bottom topography. For real water systems with complicated coastal line, the flow vector is generally not orthogonal to the tangent line for the coastline at each of its points. It also may not coincide with the wind stress vector. Therefore, it is necessary to use spatially two-dimensional models of sediment transportation and effective numerical methods to solve many practically important problems associated with the prediction of bottom surface dynamics. Materials and Methods. The spatially two-dimensional model of sediment transport that satisfies the basic conservation laws (of material balance and momentum), which is a quasilinear parabolic equation, was earlier proposed by the authors (A.I. Sukhinov, A.E. Chistyakov, E.A. Protsenko, and V.V. Sidoryakina). The linear difference schemes were constructed and researched; the model and some practically important problems were solved. However, the theoretical research of the proximity of solutions for the original nonlinear initial-boundary value problem and the linearized continuous problem, on which basis a discrete model (difference scheme) was developed, remained in the shadow. The researching correctness of the linearized problem and the determination of sufficient conditions for positivity of solutions are caused special interest because only positive solutions of this sediment transport problem have physical value within the framework of the considered models. Research Results. The investigated nonlinear two-dimensional model of sediment transport in the coastal zone of shallow water basins takes into account the following physically significant conditions and parameters: bottom material porosity; critical value of the tangent stress at which bottom material transport is started; turbulent mixing; the dynamically varying bottom geometry; wind currents; and bottom friction. Linearization is carried out on the time grid; nonlinear coefficients of the parabolic equation are taken at the previous step of time grid. Then, a set of problems, connected by the initial data, are solved; final solutions of the linearized initial boundary value problems chain on a uniform time grid were constructed, and thus, the linearization of the initial 2D nonlinear model is carried out in total time interval. Earlier, the authors proved the existence and uniqueness of the linear problem solution. A priori proximity estimates for the solutions of linearising sequence of boundary value problems and initial non-linear task have been also obtained. Conditions of its positive solution and convergence to the nonlinear sediment transport problem are defined in the norm of the Hilbert space L1 with the rate O(τ), where the τ is a time step. Discussion and Conclusions. The obtained research results of the spatially two-dimensional nonlinear sediment transport model can be used for predicting the nonlinear hydrodynamic processes, improving their accuracy and reliability due to the availability of new accounting functionality of physically important factors, including the specification of the boundary conditions.

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