Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Application of method of collocation on lines for solving nonlinear hyperbolic problems

Author: E. N. Houstis
Journal: Math. Comp. 31 (1977), 443-456
MSC: Primary 65N35
MathSciNet review: 0443379
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A collocation on lines procedure based on piecewise polynomials is applied to initial/boundary value problems for nonlinear hyperbolic partial differential equations. Optimal order a priori estimates are obtained for the error of approximation. The Crank-Nicholson discretization in time is studied and convergence rates of the collocation-Crank-Nicholson procedure are established. Finally, the superconvergence is verified at particular points for linear hyperbolic problems.

References [Enhancements On Off] (What's this?)

  • [1] I. S. BEREZIN & N. P. ZIDKOV, Computing Methods, Vols. I, II, Fizmatgiz, Moscow, 1962; English transl., Addison-Wesley, Reading, Mass.; Pergamon Press, New York, 1965. MR 22 #12685; 30 #4372. MR 0174165 (30:4372)
  • [2] E. A. CODDINGTON, An Introduction to Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1961. MR 23 #A3869. MR 0126573 (23:A3869)
  • [3] JIM DOUGLAS, JR. & TODD DUPONT, "A finite element collocation method for quasilinear parabolic equations," Math. Comp., v. 27, 1973, pp. 17-28. MR 49 #4266. MR 0339508 (49:4266)
  • [4] JIM DOUGLAS, JR. & TODD DUPONT, "A super convergence result for the approximate solution of the heat equation by a collocation method," Mathematical Foundations of Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), Academic Press, New York, 1972. MR 0373329 (51:9529)
  • [5] JIM DOUGLAS, JR. & TODD DUPONT, Collocation Methods for Parabolic Equations in a Single Space Variable (Based on $ {C^1}$-Piecewise-Polynomial Spaces), Springer Lecture Notes in Math., Vol. 385, Springer-Verlag, Berlin and New York, 1974. MR 0483559 (58:3551)
  • [6] E. N. HOUSTIS, Finite Element Methods for Solving Initial/Boundary Value Problems, Doctoral thesis, Purdue University, 1974.
  • [7] L. V. KANTOROVIČ, "Sur une méthode de resolution approchée d'equations différentielles aux derivées partielles," C. R. Acad. (Dokl.) Sci. URSS , v. 2, 1934, pp. 532-536. (Russian)
  • [8] È. B. KARPILOVSKAJA, "Convergence of a collocation method for certain boundaryvalue problems of mathematical physics," Sibirsk. Mat. Ž., v. 4, 1963, pp. 632-640. (Russian) MR 27 #6402. MR 0156479 (27:6402)
  • [9] M. H. SCHULTZ, Spline Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1973. MR 50 #15270. MR 0362832 (50:15270)
  • [10] Yu. P. YARTSEV, "Convergence of the collocation method on lines," Differencial'nye Uravnenija, v. 3, 1967, pp. 1606-1613 = Differential Equations, v. 3, 1967, pp. 838-842. MR 0221768 (36:4820)
  • [11] Yu. P. YARTSEV, "The method of line collocation," Differencial'nye Uravnenija, v. 4, 1968, pp. 925-932 = Differential Equations, v. 4, 1968, pp. 481-485.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N35

Retrieve articles in all journals with MSC: 65N35

Additional Information

Keywords: Collocation on lines method, nonlinear hyperbolic problems
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society