Deciphering singularities by discrete methods
HTML articles powered by AMS MathViewer
- by Yves Tourigny and Michael Grinfeld PDF
- Math. Comp. 62 (1994), 155-169 Request permission
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.References
-
G. A. Baker and P. Graves-Morris, Padé approximants. vol. 1, Addison-Wesley, Reading, MA, 1981.
- Marsha Berger and Robert V. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math. 41 (1988), no. 6, 841–863. MR 948774, DOI 10.1002/cpa.3160410606
- R. P. Brent and H. T. Kung, Fast algorithms for manipulating formal power series, J. Assoc. Comput. Mach. 25 (1978), no. 4, 581–595. MR 520733, DOI 10.1145/322092.322099
- J. C. Butcher, The numerical analysis of ordinary differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR 878564
- Y. F. Chang and G. Corliss, Ratio-like and recurrence relation tests for convergence of series, J. Inst. Math. Appl. 25 (1980), no. 4, 349–359. MR 578082
- George Corliss, On computing Darboux type series analyses, Nonlinear Anal. 7 (1983), no. 11, 1247–1253. MR 721410, DOI 10.1016/0362-546X(83)90056-1
- George F. Corliss, Integrating ODEs in the complex plane—pole vaulting, Math. Comp. 35 (1980), no. 152, 1181–1189. MR 583495, DOI 10.1090/S0025-5718-1980-0583495-8
- George Corliss and Y. F. Chang, Solving ordinary differential equations using Taylor series, ACM Trans. Math. Software 8 (1982), no. 2, 114–144. MR 661124, DOI 10.1145/355993.355995 J. W. Dold, Analysis of the early stage of thermal runaway, Quart. J. Mech. Appl. Math. 38 (1985), 361-387.
- Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162
- F. B. Hildebrand, Introduction to numerical analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. MR 0075670
- Einar Hille, Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 0499382 C. Hunter, personal communication, 1992.
- C. Hunter and B. Guerrieri, Deducing the properties of singularities of functions from their Taylor series coefficients, SIAM J. Appl. Math. 39 (1980), no. 2, 248–263. MR 588498, DOI 10.1137/0139022
- Yoshifumi Kimura, Parametric motion of complex-time singularity toward real collapse, Phys. D 46 (1990), no. 3, 439–448. MR 1081692, DOI 10.1016/0167-2789(90)90104-W
- J. D. Lambert and B. Shaw, A method for the numerical solution of $y^{\prime } =f(x,\,y)$ based on a self-adjusting non-polynomial interpolant, Math. Comp. 20 (1966), 11–20. MR 189252, DOI 10.1090/S0025-5718-1966-0189252-1
- Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, DOI 10.1137/1032046
- J. N. Lyness, Differentiation formulas for analytic functions, Math. Comp. 22 (1968), 352–362. MR 230468, DOI 10.1090/S0025-5718-1968-0230468-5
- Daniel I. Meiron, Gregory R. Baker, and Steven A. Orszag, Analytic structure of vortex sheet dynamics. I. Kelvin-Helmholtz instability, J. Fluid Mech. 114 (1982), 283–298. MR 647268, DOI 10.1017/S0022112082000159
- Tomoyasu Nakagawa, Blowing up of a finite difference solution to $u_{t}=u_{xx}+u_{2}.$, Appl. Math. Optim. 2 (1975/76), no. 4, 337–350. MR 423823, DOI 10.1007/BF01448176
- L. E. Payne, Improperly posed problems in partial differential equations, Regional Conference Series in Applied Mathematics, No. 22, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0463736
- Brian Shaw, Modified multistep methods based on a nonpolynomial interpolant, J. Assoc. Comput. Mach. 14 (1967), 143–154. MR 213029, DOI 10.1145/321371.321382
- Catherine Sulem, Pierre-Louis Sulem, and Hélène Frisch, Tracing complex singularities with spectral methods, J. Comput. Phys. 50 (1983), no. 1, 138–161. MR 702063, DOI 10.1016/0021-9991(83)90045-1
- Y. Tourigny and J. M. Sanz-Serna, The numerical study of blowup with application to a nonlinear Schrödinger equation, J. Comput. Phys. 102 (1992), no. 2, 407–416. MR 1187698, DOI 10.1016/0021-9991(92)90382-9
Additional Information
- © Copyright 1994 American Mathematical Society
- 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