TY - JOUR
T1 - A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 475
EP - 498
PY - 2010
DA - 2010/03
SN - 3
DO - http://doi.org/10.4208/nmtma.2010.m88051
UR - https://global-sci.org/intro/article_detail/nmtma/6010.html
KW - Finite difference method, nonlinear wave equation, blow-up.
AB - <p style="text-align: justify;">We consider a finite difference scheme for a nonlinear wave equation, whose
solutions may lose their smoothness in finite time, i.e., blow up in finite
time. In order to numerically reproduce blow-up solutions, we propose
a rule for a time-stepping, which is a variant of what was successfully
used in the case of nonlinear parabolic equations.
A &nbsp;numerical blow-up time is defined and is proved to converge, under a certain
hypothesis, to the real blow-up time as the grid size tends to zero.</p>