NUMERICAL BLOW-UP TIME FOR A PARABOLIC EQUATION WITH NONLINEAR BOUNDARY CONDITIONS

摘要

In this paper, we study numerical approximations for positive solutions of a heat equation, u_t = u_(xx), in a bounded interval, (0, 1), with a nonlinear flux boundary condition at the boundary, u_x(0, t) = 0, u_x(1, t) = u~p(1, t) which implies that the solutions blow up in finite time. By a semidiscretization using finite difference method in the space variable, we get a system of ordinary differential equations which is expected to be an approximation of the original problem. We prove that every numerical solution blows up in finite time and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero under certain assumptions. Finally, we give some numerical results to illustrate certain points of our work.