Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Numerical methods based on additive splittings for hyperbolic partial differential equations


Authors: Randall J. LeVeque and Joseph Oliger
Journal: Math. Comp. 40 (1983), 469-497
MSC: Primary 65M05
DOI: https://doi.org/10.1090/S0025-5718-1983-0689466-8
MathSciNet review: 689466
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds. These techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude. Our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales, some of which are sometimes treated exactly. The efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator. This is analyzed and a technique is discussed for reducing this error through a simple change of variables. A procedure for generating the appropriate boundary data for the intermediate solutions is also presented.


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

  • [1] S. Abarbanel & D. Gottlieb, Optimal Time Splitting for Two and Three Dimensional Navier-Stokes Equations With Mixed Derivatives, ICASE Report No. 80-6, 1980. MR 624738 (82f:65097)
  • [2] G. Browning, A. Kasahara & H.-O. Kreiss, "Initialization of the primitive equations by the bounded derivative method," J. Atmospheric Sci., v. 37, 1980, pp. 1424-1436. MR 620275 (82e:65125)
  • [3] J. Certaine, "The solution of ordinary differential equations with large time constants," in Mathematical Methods for Digital Computers (A. Ralston and H. S. Wilf, eds.), Wiley, New York, 1960, pp. 128-132. MR 0117917 (22:8691)
  • [4] B. Engquist, B. Gustafsson & J. Vreeburg, "Numerical solution of a PDE system describing a catalytic converter," J. Comput. Phys., v. 27, 1978, pp. 295-314. MR 0488887 (58:8385)
  • [5] A. J. Gadd, "A split explicit integration scheme for numerical weather prediction," Quart. J. Roy. Met. Soc., v. 104, 1978, pp. 569-582.
  • [6] A. R. Gourlay, "Splitting methods for time dependent partial differential equations," in The State of the Art in Numerical Analysis (D. Jacobs, ed.), Academic Press, New York, 1977. MR 0451759 (56:10041)
  • [7] B. Gustafsson, "The convergence rate for difference approximations to mixed initial boundary value problems," Math. comp., v. 29, 1975, pp. 396-406. MR 0386296 (52:7154)
  • [8] 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. 649-685. MR 0341888 (49:6634)
  • [9] H.-O. Kreiss, "Problems on different time scales for ordinary differential equations," SIAM J. Numer. Anal., v. 16, 1979, pp. 980-998. MR 551320 (81a:65087)
  • [10] H.-O. Kreiss, "Problems with different time scales for partial differential equations," Comm. Pure Appl. Math., v. 33, 1980, pp. 399-439. MR 562742 (81c:35082)
  • [11] J. D. Lawson & J. Ll. Morris, A Review of Splitting Methods, Report CS-74-09, Department of Applied Analysis and Computer Science, University of Waterloo, 1974.
  • [12] G. Majda, Filtering Techniques for Oscillatory Stiff Ordinary Differential Equations, Thesis, New York University.
  • [13] A. R. Mitchell, Computational Methods in Partial Differential Equations, Wiley, New York, 1969. MR 0281366 (43:7084)
  • [14] R. E. O'Malley & L. R. Anderson, "Singular perturbations, order reduction, and decoupling of large scale systems," in Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, eds.), Academic Press, New York, 1979, pp. 317-998. MR 556524 (80m:34050)
  • [15] R. D. Richtmyer & K. W. Morton, Difference Methods for Initial-Value Problems, Interscience Tracts in Pure and Appl. Math., No. 4, Wiley, New York, 1967. MR 0220455 (36:3515)
  • [16] G. Strang, "On the construction and comparison of difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 506-517. MR 0235754 (38:4057)
  • [17] J. Strikwerda, A Time-Split Difference Scheme for the Compressible Navier-Stokes Equations With Applications to Flows in Slotted Nozzles, ICASE Report No. 80-27, 1980.
  • [18] E. Turkel & G. Zwas, "Explicit large time-step schemes for the shallow water equations," AICA Proc., No. 3, 1979, pp. 65-69.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M05

Retrieve articles in all journals with MSC: 65M05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1983-0689466-8
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society