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

TL;DR

This paper presents an efficient method for approximating positive stationary solutions of a discretized semilinear heat equation with small absorption, using a homotopy continuation algorithm with linear complexity.

Contribution

It introduces a homotopy continuation algorithm that computes epsilon-approximations of solutions with linear cost relative to discretization size and precision.

Findings

Unique solution exists for small absorption levels.

Algorithm has linear complexity in the number of discretization points.

Approximations can be computed efficiently with controlled accuracy.

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.