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Volume 3, Issue 4
A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations

Chien-Hong Cho

DOI: 10.4208/nmtma.2010.m88051

Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 475-498.

Published online: 2010-03

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  • Abstract

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  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.

  • Keywords

Finite difference method, nonlinear wave equation, blow-up.

  • AMS Subject Headings

65M06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-3-475, author = {Chien-Hong Cho}, title = {A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2010}, volume = {3}, number = {4}, pages = {475--498}, abstract = {

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  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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2010.m88051}, url = {http://global-sci.org/intro/article_detail/nmtma/6010.html} }
TY - JOUR T1 - A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations AU - Chien-Hong Cho 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 -

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  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.

Chien-Hong Cho. (2010). A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations. Numerical Mathematics: Theory, Methods and Applications. 3 (4). 475-498. doi:10.4208/nmtma.2010.m88051
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