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


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


Zbl 0632.35028
Full Text: DOI


[1] and , Complete blow-up after Tmix for the solution of a semilinear heat equation, to appear.
[2] Bebernes, Ann. Inst. H. Poincaré-Analyse Nonlineaire 5 pp 1– (1988)
[3] Berger, Math. Comp. 45 pp 301– (1985)
[4] Berger, J. Comp. Phys. 53 pp 484– (1984)
[5] On blow-up solutions of semilinear parabolic equations; analytical and numerical studies, Thesis, Tokyo University, 1987,.
[6] Chen, J. Fac. Sci. Univ. Tokyo Sect. IA (Math) 33 pp 541– (1986)
[7] Chorin, Comm. Pure Appl. Math. 34 pp 853– (1981)
[8] Ciment, Math. Comp. 25 pp 219– (1971)
[9] Dold, Q. J. Mech. Appl. Math. 38 pp 361– (1985)
[10] Blowup of solutions of nonlinear parabolic equations, in Proc. of Microprogram on Nonlinear Diffusion Equations and their Equilibrium States, MSRI, Berkeley, 1986, to appear.
[11] Friedman, Indiana Univ. Math. J. 34 pp 425– (1985)
[12] Fujita, Analyse Math, et Appl.
[13] and , The equation ut = uxx + u{\(\beta\)}. Localization and asymptotic behavior of unbounded solutions, Preprint no. 97, Keldysh Institute of Applied Math., Moscow, 1985,. (In Russian.)
[14] Galaktionov, Differ. Uravnen. 22 pp 1165– (1986)
[15] Self-similar solutions for semilinear parabolic equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, B. Nicolaenko et al. eds., AMS Lecture Notes in Appl. Math. 23, 1986, Part 2, pp. 247–252.
[16] Giga, Comm. Pure Appl. Math. 38 pp 297– (1985)
[17] Giga, Indiana Univ. Math. J. 36 pp 1– (1987)
[18] and , Removability of blow-up points for semilinear heat equations, to appear in proc. EQUADIFF, ed., Marcel Dekker, 1987.
[19] and , Nondegeneracy of blow-up for semilinear heat equations, preprint. · Zbl 0703.35020
[20] Hocking, J. Fluid Mech. 51 pp 705– (1972)
[21] Kapila, SIAM J. Appl. Math. 39 pp 21– (1980)
[22] Kassoy, SIAM J. Appl. Math. 39 pp 412– (1980)
[23] The form of blowup for nonlinear parabolic equations, Proc. Roy. Soc. Edinburgh 98A, 1984, pp. 193–202.
[24] , , and , The focusing singularity of the nonlinear Schrödinger equation, in Directions in Partial Differential Equations, et al. eds., Academic Press, 1987, pp. 159–201. · doi:10.1016/B978-0-12-195255-6.50016-7
[25] LeMesurier, Phys. Rev. A34 pp 1200– (1986)
[26] Mueller, Indiana Univ. Math. J. 34 pp 881– (1985)
[27] Sulem, Comm. Pure Appl. Math. 37 pp 755– (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.