Article
Full entry |
PDF
(1.1 MB)
Feedback
PDF
(1.1 MB)
Feedback
Keywords:
noncoercive nonsymmetric problems; Helmholtz equation; finite difference; alternating-direction iteration method; time-stepping method; convergence; numerical examples
noncoercive nonsymmetric problems; Helmholtz equation; finite difference; alternating-direction iteration method; time-stepping method; convergence; numerical examples
Summary:
An alternating-direction iterative procedure is described for a class of Helmholz-like problems. An algorithm for the selection of the iteration parameters is derived; the parameters are complex with some having positive real part and some negative, reflecting the noncoercivity and nonsymmetry of the finite element or finite difference matrix. Examples are presented, with an applications to wave propagation.
References:
[1] Douglas J., Jr.: On the numerical integration of $u_{xx} + u_{yy} = u_t$ by implicit methods. J. Soc. Indust. Appl. Math. 3(1955), 42-65. MR 0071875
[2] Douglas J., Jr.: Alternating direction methods for three space variables. Numerische Mathematik 4 (1962), 41-63. DOI 10.1007/BF01386295 | MR 0136083 | Zbl 0104.35001
[3] Douglas J., Jr., Dupont T.: Alternating-direction Galerkin methods on rectangles. Numerical Solution of Partial Differential Equations II (Burt Hubbard, ed.), Academic Press, New York, 1971, pp. 133-214. MR 0273830 | Zbl 0239.65088
[4] Douglas J., Jr., Gunn J. E.: A general formulation of alternating direction methods, I. Parabolic and hyperbolic problems. Numerische Mathematik 6 (1964), 428-453. DOI 10.1007/BF01386093 | MR 0176622 | Zbl 0141.33103
[5] Douglas J., Jr., Peaceman D. W.: Numerical solution of two dimensional heat flow problems. A.I.Ch.E. Jour 1 (1955), 505-512.
[6] Douglas J., Jr., Rachford H. H., Jr.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Soc. 82 (1956), 421-439. DOI 10.1090/S0002-9947-1956-0084194-4 | MR 0084194 | Zbl 0070.35401
[7] Douglas J., Jr., Santos J. E., Sheen D., Bennethum L. S.: Frequency domain treatment of one-dimensional scalar waves. Mathematical Models and Methods in Applied Sciencis (1993), to appear. MR 1212938 | Zbl 0783.65070
[8] Peaceman D. W.: The numerical solution of parabolic elliptic differential equations. J. Soc. Ind. Appl. Math. 3 (1955), 28-41. DOI 10.1137/0103003 | MR 0071874
[9] Pearcy C. M.: On convergence of alternating direction procedures. Numerische Mathematik 4 (1962), 172-176. DOI 10.1007/BF01386310 | MR 0145677 | Zbl 0112.34802
Search
Partner of
