Numerical solution of the cross-diffusion problem with nonlinear boundary conditions and source
Keywords:
asymptotics, cross-diffusion, nonlinear system, self-similar solution, iterationAbstract
This paper investigates the qualitative properties of self-similar solutions of a nonlinear cross-diffusion system with nonlinear boundary conditions and a source. A self-similar solution of the cross-diffusion problem is constructed. In the case of global solvability, the leading term of the asymptotics of self-similar solutions of the cross-diffusion problem with a source is obtained. When numerically solving nonlinear problems using iterative methods, it is crucial to choose a suitable initial approximation that preserves the nonlinear properties. Computational experiments have shown the rapid convergence of the iterative process to the exact solution, due to the choice of a suitable initial approximation. The use of asymptotic formulas as initial approximations in iterative methods significantly increases the chances of rapid convergence of the algorithm and improves the accuracy of the numerical results obtained.
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