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Zbl 0652.65070
Berger, Marsha; Kohn, Robert V.
A rescaling algorithm for the numerical calculation of blowing-up solutions. (English)
[J] Commun. Pure Appl. Math. 41, No.6, 841-863 (1988). ISSN 0010-3640

A method is developed for computing solutions of certain nonlinear evolution equations near a developing singularity in space-time. The main tools are rescaling and mesh refinement; in essence, the method uses a varying spatial grid and time step, linked at each point of space-time to the magnitude of the computed solution. The discussion is focussed on the specific equation $u\sb t-u\sb{xx}=u\quad p$ on an interval, with $u=0$ at the endpoints. The numerical results, which remain accurate as the magnitude of u grows from O 1 to $O(10\sp{12}),$, agree with the behavior conjectured by {\it V. A. Galaktionov} and {\it S. A. Posashkov} [Diff. Uravn. 22, 1165-1173 (1986; Zbl 0632.35028)] on the basis of a formal expansion.
[M.Berger]

MSC 2000:: *65N06 Finite difference methods (BVP of PDE)
35K60 (Nonlinear) BVP for (non)linear parabolic equations
35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general

Keywords: blowing-up solutions; nonlinear evolution equations; rescaling; mesh refinement

Citations: Zbl 0632.35028

Cited in: Zbl 0872.35049

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