Efficient approximation of the solution of certain nonlinear reaction--diffusion equation II: the case of large absorption

TL;DR

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

Contribution

It introduces a novel linear-cost algorithm for approximating solutions of discretized nonlinear reaction-diffusion equations with large absorption.

Findings

Unique solution exists for large enough absorption.

Algorithm computes epsilon-approximations efficiently.

Cost is linear in discretization size and logarithmic in precision.

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 large 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.