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

2011/03/02 by Ezequiel Dratman, Dratman, Ezequiel
Computer Science · Engineering · Mathematics · #65H10 #65H20 #65L10 #65L12 #65Y20 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1103.0491

openalex publication_date 2011/03/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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 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 ε-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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