Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Fourth order difference methods for the initial boundary-value problem for hyperbolic equations

Author: Joseph Oliger
Journal: Math. Comp. 28 (1974), 15-25
MSC: Primary 65N05
MathSciNet review: 0359344
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Centered difference approximations of fourth order in space and second order in time are applied to the mixed initial boundary-value problem for the hyperbolic equation $ {u_t} = - c{u_x}$. A method utilizing third order uncentered differences at the boundaries is shown to be stable and to retain an overall fourth order convergence estimate. Several computational examples illustrate the success of these methods for problems with one and two spacial dimensions. Further examples illustrate the effects of approximations of various orders of accuracy used at the boundaries.

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

  • [1] G. E. Collins, "The calculation of multivariate polynomial resultants," J. Assoc. Comput. Mach., v. 18, 1971, pp. 515-532. MR 45 #7970. MR 0298921 (45:7970)
  • [2] G. E. Collins, The SAC-1 Polynomial GCD and Resultant System, Report 145, Computer Sciences Department, University of Wisconsin, Madison, 1972.
  • [3] T. Elvius & A. Sundström, "Computationally efficient schemes and boundary conditions for a fine-mesh barotropic model based on the shallow-water equations," Tellus, v. 25, 1973, pp. 132-156.
  • [4] B. Fornberg, On High Order Approximations of Hyperbolic Partial Differential Equations by a Fourier Method, Report 39, Department of Computer Sciences, Uppsala University, Uppsala, Sweden, 1972.
  • [5] B. Gustafsson, On the Convergence Rate for Difference Approximations to Mixed Initial Boundary Value Problems, Report 33, Department of Computer Sciences, Uppsala University, Uppsala, Sweden, 1971.
  • [6] B. Gustafsson, H.-O. Kreiss & A. Sundström, "Stability theory of difference approximations for mixed initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649-686. MR 0341888 (49:6634)
  • [7] M. A. Jenkins & J. F. Traub, "A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration," Numer. Math., v. 14, 1969/70, pp. 252-263. MR 41 #2918. MR 0258271 (41:2918)
  • [8] H.-O. Kreiss, "Difference approximations for initial boundary value problems," Proc. Roy. Soc. London Ser. A, v. 323, 1971, pp. 255-261. MR 0501979 (58:19187)
  • [9] H.-O. Kreiss & J. Oliger, "Comparison of accurate methods for the integration of hyperbolic equations," Tellus, v. 24, 1972, pp. 199-215. MR 0319382 (47:7926)
  • [10] M. Marden, Geometry of Polynomials, 2nd ed., Math. Surveys, no. 3, Amer. Math. Soc., Providence, R.I., 1966. MR 37 #1562. MR 0225972 (37:1562)
  • [11] J. J. H. Miller, "On the location of zeros of certain classes of polynomials with applications to numerical analysis," J. Inst. Math. Appl., v. 8, 1971, pp. 397-406. MR 45 #9481. MR 0300435 (45:9481)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N05

Retrieve articles in all journals with MSC: 65N05

Additional Information

Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society