Deciphering singularities by discrete methods

Authors:
Yves Tourigny and Michael Grinfeld

Journal:
Math. Comp. **62** (1994), 155-169

MSC:
Primary 65L05; Secondary 65P05

DOI:
https://doi.org/10.1090/S0025-5718-1994-1203737-5

MathSciNet review:
1203737

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Abstract: We consider the problem of estimating numerically the parameters of singularities of solutions of differential equations. We propose a novel approach which is based on discretizing the governing equation and "time-stepping" in the complex domain. Some applications to ordinary and partial differential equations are discussed.

**[1]**G. A. Baker and P. Graves-Morris,*Padé approximants*. vol. 1, Addison-Wesley, Reading, MA, 1981.**[2]**M. Berger and R. V. Kohn,*A rescaling algorithm for the numerical calculation of blowing-up solutions*, Comm. Pure Appl. Math.**41**(1988), 841-863. MR**948774 (89g:65154)****[3]**R. P. Brent and H. T. Kung,*Fast algorithms for manipulating formal power series*, J. Assoc. Comput. Mach.**25**(1978), 581-595. MR**0520733 (58:25090)****[4]**J. C. Butcher,*The numerical analysis of ordinary differential equations, Runge-Kutta methods and general linear methods*, Wiley, Chichester, 1987. MR**878564 (88d:65002)****[5]**Y. F. Chang and G. Corliss,*Ratio-like and recurrence tests for convergence of series*, IMA J. Appl. Math.**25**(1980), 349-359. MR**578082 (81i:40004)****[6]**G. Corliss,*On computing Darboux type series analyses*, Nonlinear Anal.**7**(1983), 1247-1253. MR**721410 (86d:65042)****[7]**-,*Integrating ODE's in the complex plane--pole vaulting*, Math. Comp.**35**(1980), 1181-1189. MR**583495 (81m:65115)****[8]**G. Corliss and Y. F. Chang,*Solving ordinary differential equations using Taylor series*, ACM Trans. Math. Software**8**(1982), 114-144. MR**661124 (83g:65072)****[9]**J. W. Dold,*Analysis of the early stage of thermal runaway*, Quart. J. Mech. Appl. Math.**38**(1985), 361-387.**[10]**P. Henrici,*Applied and computational complex analysis*, vol. 1, Wiley, New York, 1974. MR**0372162 (51:8378)****[11]**F. Hildebrand,*Introduction to numerical analysis*, McGraw-Hill, New York, 1956. MR**0075670 (17:788d)****[12]**E. Hille,*Ordinary differential equations in the complex domain*, Wiley, New York, 1976. MR**0499382 (58:17266)****[13]**C. Hunter, personal communication, 1992.**[14]**C. Hunter and B. Guerrieri,*Deducing the properties of singularities of functions from their Taylor series coefficients*, SIAM J. Appl. Math.**39**(1980), 248-263. MR**588498 (82b:65015a)****[15]**Y. Kimura,*Parametric motion of complex-time singularity toward real collapse*, Phys. D**46**(1990), 439-448. MR**1081692 (91j:34067)****[16]**J. D. Lambert and B. Shaw,*A method for the numerical solution of**based on a selfadjusting non-polynomial interpolant*, Math. Comp.**20**(1966), 11-20. MR**0189252 (32:6679)****[17]**H. A. Levine,*The role of critical exponents in blowup theorems*, SIAM Rev.**32**(1990), 262-288. MR**1056055 (91j:35135)****[18]**J. N. Lyness,*Differentiation formulas for analytic functions*, Math Comp.**21**(1967), 352-362. MR**0230468 (37:6030)****[19]**D. I. Meiron, G. R. Baker, and S. Orszag,*Analytic structure of vortex sheet dynamics*. Part 1.*Kelvin-Helmholtz instability*, J. Fluid Mech.**114**(1982), 283-298. MR**647268 (83c:76017)****[20]**T. Nakagawa,*Blowing up of a finite difference solution to*, Appl. Math. Optim.**2**(1976), 337-350. MR**0423823 (54:11797)****[21]**L. E. Payne,*Improperly posed problems in partial differential equations*, SIAM, Philadelphia, PA, 1975. MR**0463736 (57:3678)****[22]**B. Shaw,*Modified multistep methods based on a nonpolynomial interpolant*, J. Assoc. Comput. Mach.**14**(1967), 143-154. MR**0213029 (35:3894)****[23]**C. Sulem, P. L. Sulem, and H. Frisch,*Tracing complex singularities with spectral methods*, J. Comput. Phys.**50**(1983), 138-161. MR**702063 (84m:58088)****[24]**Y. Tourigny and J. M. Sanz-Serna,*The numerical study of blowup with application to a nonlinear Schrödinger equation*, J. Comput. Phys.**102**(1992). MR**1187698 (93g:65161)**

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DOI:
https://doi.org/10.1090/S0025-5718-1994-1203737-5

Article copyright:
© Copyright 1994
American Mathematical Society