One-step piecewise polynomial multiple collocation methods for initial value problems

Author:
J. P. Hennart

Journal:
Math. Comp. **31** (1977), 24-36

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1977-0431686-9

MathSciNet review:
0431686

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Abstract | References | Similar Articles | Additional Information

Abstract: New methods are proposed for the numerical solution of systems of first-order differential equations. On each subinterval of a given mesh of size *h*, a polynomial of degree *l* is constructed, its parameters being determined by a multiple collocation technique. The resulting piecewise polynomial approximation is of order at the mesh points and between them. In addition, the *j*th derivatives of the approximation on each subinterval provide approximations of order , . Some of the methods proposed are shown to be *A*-stable or even strongly *A*-stable.

**[1]**G. D. ANDRIA, G. D. BYRNE & D. R. HILL, "Natural spline block implicit methods,"*BIT*, v. 13, 1973, pp. 131-144. MR**48**#1468. MR**0323110 (48:1468)****[2]**G. D. ANDRIA, G. D. BYRNE & D. R. HILL, "Integration formulas and schemes based on*g*-splines,"*Math. Comp.*, v. 27, 1973, pp. 831-838; addendum,*ibid.*, microfiche supplement A1-C4. MR**49**#4219. MR**0339460 (49:4219)****[3]**J. H. ARGYRIS & D. W. SCHARPF, "Finite elements in time and space,"*J. Royal Aeronautical Soc.*, v. 73, 1969, pp. 1041-1044.**[4]**G. BIRKHOFF & R. S. VARGA, "Discretization errors for well-set Cauchy problems. I,"*J. Math. and Phys.*, v. 44, 1965, pp. 1-23. MR**31**#4189. MR**0179952 (31:4189)****[5]**J. C. BRUCH & G. ZYVOLOSKI, "Finite element solution of unsteady and unsaturated flow in porous media,"*The Mathematics of Finite Elements and Applications*, J. R. Whiteman (Editor), Academic Press, London and New York, 1973, pp. 201-211.**[6]**G. D. BYRNE & D. N. H. CHI, "Linear multistep formulas based on*g*-splines,"*SIAM J. Numer. Anal.*, v. 9, 1972, pp. 316-324. MR**46**#10207. MR**0311111 (46:10207)****[7]**E. D. CALLENDER, "Single step methods and low order splines for solutions of ordinary differential equations,"*SIAM J. Numer. Anal.*, v. 8, 1971, pp. 61-66. MR**47**#4446. MR**0315897 (47:4446)****[8]**P. J. DAVIS,*Interpolation and Approximation*, Blaisdell, New York, 1965. MR**0157156 (28:393)****[9]**G. G. DAHLQUIST, "A special stability problem for linear multistep methods,"*BIT*, v. 3, 1963, pp. 27-43. MR**30**#715. MR**0170477 (30:715)****[10]**J. DOUGLAS, JR. & T. DUPONT, "Galerkin methods for parabolic equations",*SIAM J. Numer. Anal.*, v. 7, 1970, pp. 575-626. MR**43**#2863. MR**0277126 (43:2863)****[11]**B. L. EHLE, "*A*-stable methods and Padé approximations to the exponential,"*SIAM J. Math. Anal.*, v. 4, 1973, pp. 671-680. MR**48**#10119. MR**0331787 (48:10119)****[12]**G. FIX & N. NASSIF, "On finite element approximations to time-dependent problems,"*Numer. Math.*, v. 19, 1972, pp. 127-135. MR**46**#10218. MR**0311122 (46:10218)****[13]**J. P. HENNART, "Piecewise polynomials for point and space kinetics with variable reactivity,"*Trans. Amer. Nuclear Soc.*, v. 19, 1974, pp. 179-180.**[14]**J. P. HENNART, "Piecewise polynomial multiple collocation methods for initial value problems," Notas de Matemática y Simposia, v. 2, Sociedad Matemática Mexicana. (To appear.)**[15]**P. HENRICI,*Discrete Variable Methods in Ordinary Differential Equations*, Wiley, New York, 1962. MR**24**#B1772. MR**0135729 (24:B1772)****[16]**B. L. HULME, "Piecewise polynomial Taylor methods for initial value problems,"*Numer. Math.*, v. 17, 1971, pp. 367-381. MR**45**#8002. MR**0298953 (45:8002)****[17]**B. L. HULME, "One-step piecewise polynomial Galerkin methods for initial value problems,"*Math. Comp.*, v. 26, 1972, pp. 415-426. MR**47**#9834. MR**0321301 (47:9834)****[18]**B. L. HULME, "Discrete Galerkin and related one-step methods for ordinary differential equations,"*Math. Comp.*, v. 26, 1972, pp. 881-891. MR**47**#4448. MR**0315899 (47:4448)****[19]**P. M. HUMMEL & C. L. SEEBECK, JR., "A generalization of Taylor's expansion,"*Amer. Math. Monthly*, v. 56, 1949, pp. 243-247. MR**10**, 516. MR**0028907 (10:516i)****[20]**C. M. KANG & K. F. HANSEN, "Finite element methods for reactor analysis,"*Nuclear Sci. and Engrg.*, v. 51, 1973, pp. 456-495.**[21]**F. R. LOSCALZO & T. D. TALBOT, "Spline function approximations for solutions of ordinary differential equations,"*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 433-445. MR**36**#4808. MR**0221756 (36:4808)****[22]**F. R. LOSCALZO, "An introduction to the application of spline functions to initial value problems,"*Theory and Applications of Spline Functions*(Proc. Seminar, Math. Res. Center, Univ. of Wisconsin, 1968), T.N.E. Greville (Editor), Academic Press, New York, 1969, pp. 37-64. MR**39**#2334. MR**0240989 (39:2334)****[23]**J. T. ODEN, "A general theory of finite elements. Part II, Applications,"*Internat. J. Numer. Methods Engrg.*, v. 1, 1969, pp. 247-259.**[24]**H. S. PRICE & R. S. VARGA, "Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics,"*Numerical Solutions of Field Problems in Continuum Physics*(Proc. Sympos. Appl. Math., Durham, N. C., 1968), G. Birkhoff and R. S. Varga (Editors), SIAM-AMS Proc., Vol. II, Amer. Math. Soc., Providence, R. I., 1970, pp. 74-94. MR**42**#1358. MR**0266452 (42:1358)****[25]**B. SWARTZ & B. WENDROFF, "Generalized finite-difference schemes,"*Math. Comp.*, v. 23, 1969, pp. 37-49. MR**39**#1125. MR**0239768 (39:1125)****[26]**R. S. VARGA, "On higher order stable implicit methods for solving parabolic partial differential equations,"*J. Math. and Phys.*, v. 40, 1961, pp. 220-231. MR**25**#3613. MR**0140191 (25:3613)****[27]**G. WARZEE, "Finite element analysis of transient heat conduction. Application of the weighted residual process,"*Comput. Methods Appl. Mech. Engrg.*, v. 3, 1974, pp. 255-268.**[28]**O. B. WIDLUND, "A note on unconditionally stable linear multistep methods,"*BIT*, v. 7, 1967, pp. 65-70. MR**35**#6373. MR**0215533 (35:6373)****[29]**H. A. WRIGHT, "Some relationship between implicit Runge-Kutta, collocation and Lanczos methods, and their stability properties,"*BIT*, v. 10, 1970, pp. 217-227. MR**0266439 (42:1345)****[30]**O. C. ZIENKIEWICZ & C. J. PAREKH, "Transient field problems two and three dimensional analysis by isoparametric finite elements,"*Internat. J. Numer. Methods Engrg.*, v. 2, 1970, pp. 61-71.**[31]**O. C. ZIENKIEWICZ,*The Finite Element Method in Engineering Science*, McGraw-Hill, London and New York, 1971. MR**47**#4518. MR**0315970 (47:4518)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0431686-9

Keywords:
Initial value problems,
ordinary differential equations,
piecewise polynomials,
collocation methods,
*A*-stability

Article copyright:
© Copyright 1977
American Mathematical Society