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arXiv · 1103.0491

Efficient approximation of the solution of certain nonlinear reaction--diffusion equation I: the case of small absorption

Ezequiel Dratman
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Abstract

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if \emph{the absorption is small enough}, compared with the flux in the boundary, there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the "continuous" equation. Furthermore, we exhibit an algorithm computing an $\epsilon$-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is {\em linear} in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.

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Ezequiel Dratman. 2011-03-02. Efficient approximation of the solution of certain nonlinear reaction--diffusion equation I: the case of small absorption. https://arxiv.org/abs/1103.0491

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