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A rescaling algorithm for the numerical calculation of blowing-up solutions. (English) Zbl 0652.65070

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 \(u_ t-u_{xx}=u\quad p\) 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 \(O(10^{12}),\), agree with the behavior conjectured by V. A. Galaktionov and S. A. Posashkov [Diff. Uravn. 22, 1165-1173 (1986; Zbl 0632.35028)] on the basis of a formal expansion.
Reviewer: M.Berger

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations

Citations:

Zbl 0632.35028
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References:

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