TY - JOUR
T1 - Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-up Problem
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 679
EP - 696
PY - 2018
DA - 2018/02
SN - 7
DO - http://doi.org/10.4208/eajam.220816.300517a
UR - https://global-sci.org/intro/article_detail/eajam/10713.html
KW - Blow-up, numerical blow-up time, finite difference method, nonlinear ODE.
AB - <p style="text-align: justify;">We consider the second order nonlinear ordinary differential equation $u′′(t)=u^{1+α}(α&gt;0)$ with positive initial data $u(0)=a_0$ , $u′(0)=a_1$ , whose solution becomes
unbounded in a finite time $T$. The finite time $T$ is called the blow-up time. Since finite
difference schemes with uniform meshes can not reproduce such a phenomenon well,
adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness
has been considered before. However, more iterations are required to obtain an approximate
blow-up time if smaller meshes are applied. As a consequence, we consider in this
paper a finite difference scheme with a rather larger grid size and show the convergence
of the numerical solution and the numerical blow-up time. Application to the nonlinear
wave equation is also discussed.</p>