On the finite difference approximation for a parabolic blow-up problem

On the convergence of numerical blow-up time for a second order nonlinear ordinary differential equation

“…The blow-up time for a first order ODEẇ = f (w) was considered in [2]. Our theorem is its generalization to a second order ODE.…”

“…However, we could not derive the rate of convergence for the numerical blow-up time. The rate of convergence is not known in the parabolic blow-up problem either, see [2,7]. The only exception is the case of a scalar ODE of first order.…”

“…The conclusion of Theorem 2 itself is not important. In fact, if we wish to compute the blow-up time of (1), we simply apply the result of [2] to (2). The importance of Theorem 2 lies in its role as a first step to the nonlinear wave blow-up problem…”

“…See, for instance, [1][2][3][4][5][6][7]. Numerical estimation of the blow-up is important, but the error bound for the blow-up time is not known, except for a scalar ordinary differential equation of first order in [2].…”

“…It is known that a finite difference scheme with uniform time mesh cannot reproduce blow-up phenomenon for a blowup problem, see [2]. Thus, we consider the following scheme with initial data u 0 = a 0 > 0, and u 1 = a 0 + a 1 ∆t 0 .…”