Sufficient Conditions for the Convergence of Solutions of the Linearized Problem to the Solution of the Original Nonlinear Problem of Multifractional Sediment Transport in Shallow Water

Introduction. This paper examines a two-dimensional spatial model of multifractional sediment transport, specifically focusing on shallow water zones. This process can be described using an initial-boundary value problem for a parabolic equation with nonlinear coefficients. The study employs a temporal grid linearization method with a step size τ, where nonlinear coefficients are calculated with a “lag” at the previous time layer. Previously, the well-posedness conditions for the linearized sediment transport problem were established, and a conservative and stable finite-difference scheme was developed and analyzed, with numerical implementations for both model and real-world problems (the Sea of Azov, the Taganrog Bay, and the Tsimlyansk Reservoir). However, the convergence of solutions of the linearized problem to the solution of the original nonlinear initial-boundary value problem for multifractional sediment transport had not yet been explored. The research results presented in this paper fill this gap. Earlier, the author, together with A.I. Sukhinov, conducted similar studies in the case where sediment fraction composition was not considered. These studies formed the basis for obtaining new results.

Materials and Methods. The derivation of inequalities guaranteeing the convergence of the solutions of a sequence of linearized problems to the solution of the original nonlinear problem is carried out using the method of mathematical induction, with the application of differential equation theory.

Results. The conditions for the convergence of solutions of the linearized multifractional sediment transport problem to
the solution of the nonlinear problem in the Banach space L₁
 norm at a rate O(τ) of are determined.

Discussion and Conclusion. The obtained research results can be used for forecasting nonlinear hydrophysical processes, improving their accuracy and reliability due to the new functional capabilities that account for physically significant factors.

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