Numerical solution to the problem of cross diffusion with nonlocal boundary conditions

In this paper, we study the asymptotic behavior of self-similar solutions of a nonlinear cross-diffusion system coupled in the nonlocal boundary conditions. On the basis of selfsimilaranalysis the main term of the asymptotics of self-similar solutions is obtained. A numerical scheme is constructed based on the finite difference method. For this, equationsare approximated with the second order of accuracy in spatial coordinates and with the first order in t. An iterative process is constructed, in the inner iteration steps, the nodevalues are calculated by the sweep method. Provided a method of selecting a suitable initial approximation for the iterative process in numerical studies of the problem. Usingthe asymptotic formula as the initial approximation for the iterative process, a numerical calculation carried out and analyzed the results. Results of numerical experimentsshow that the obtained results are in good agreement with the physics of the process of nonlinear diffusion.

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