On the solution of the polynomial systems arising in the discretization of certain ODEs

Authors:
Ezequiel Dratman;Guillermo Matera
Affiliations:
Universidad Nacional de General Sarmiento, Instituto de Ciencias, Juan M. Gutiérrez 1150, B1613GSX, Los Polvorines, Buenos Aires, Argentina;Universidad Nacional de General Sarmiento, Instituto del Desarrollo Humano, Juan M. Gutiérrez 1150, B1613GSX, Los Polvorines, Buenos Aires, Argentina
Venue:
Computing
Year:
2009

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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 the diffusion is large enough, then there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the boundary-value problem under consideration. Furthermore, in this case 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 polynomial in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.