Numerical method of estimating the blow-up time and rate of the solution of ordinary differential equations—An application to the blow-up problems of partial differential equations

A numerical method is proposed for estimating the blow-up time and the blow-up rate of the solution of ordinary differential equation (ODE), when the solution diverges at a finite time, that is, the blow-up time. The main idea is to transform the ODE into a tractable form by the arc length transformation   technique [S. Moriguti, C. Okuno, R. Suekane, M. Iri, K. Takeuchi, Ikiteiru Suugaku—Suuri Kougaku no Hatten (in Japanese), Baifukan, Tokyo, 1979.], and to generate a linearly convergent sequence to the blow-up time. The sequence is then accelerated by the Aitken Δ2Δ2 method. The present method is applied to the blow-up problems of partial differential equations (PDEs) by discretising the equations in space and integrating the resulting ODEs by an ODE solver, that is, the method of lines approach. Numerical experiments on the three PDEs, the semi-linear reaction–diffusion equation, the heat equation with a nonlinear boundary condition and the semi-linear reaction–diffusion system, show the validity of the present method.