Fourth order difference methods for the initial boundaryvalue problem for hyperbolic equations
Author:
Joseph Oliger
Journal:
Math. Comp. 28 (1974), 1525
MSC:
Primary 65N05
MathSciNet review:
0359344
Fulltext 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 boundaryvalue problem for the hyperbolic equation . 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.
 [1]
G. E. Collins, "The calculation of multivariate polynomial resultants," J. Assoc. Comput. Mach., v. 18, 1971, pp. 515532. MR 45 #7970. MR 0298921 (45:7970)
 [2]
G. E. Collins, The SAC1 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 finemesh barotropic model based on the shallowwater equations," Tellus, v. 25, 1973, pp. 132156.
 [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. 649686. MR 0341888 (49:6634)
 [7]
M. A. Jenkins & J. F. Traub, "A threestage variableshift iteration for polynomial zeros and its relation to generalized Rayleigh iteration," Numer. Math., v. 14, 1969/70, pp. 252263. 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. 255261. 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. 199215. 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. 397406. MR 45 #9481. MR 0300435 (45:9481)
 [1]
 G. E. Collins, "The calculation of multivariate polynomial resultants," J. Assoc. Comput. Mach., v. 18, 1971, pp. 515532. MR 45 #7970. MR 0298921 (45:7970)
 [2]
 G. E. Collins, The SAC1 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 finemesh barotropic model based on the shallowwater equations," Tellus, v. 25, 1973, pp. 132156.
 [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. 649686. MR 0341888 (49:6634)
 [7]
 M. A. Jenkins & J. F. Traub, "A threestage variableshift iteration for polynomial zeros and its relation to generalized Rayleigh iteration," Numer. Math., v. 14, 1969/70, pp. 252263. 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. 255261. 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. 199215. 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. 397406. 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
DOI:
http://dx.doi.org/10.1090/S00255718197403593447
PII:
S 00255718(1974)03593447
Article copyright:
© Copyright 1974
American Mathematical Society
