Unconditional convergence of some Crank-Nicolson LOD methods for initial-boundary value problems

Author:
Willem Hundsdorfer

Journal:
Math. Comp. **58** (1992), 35-53

MSC:
Primary 65M12; Secondary 65M20

MathSciNet review:
1106972

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper convergence properties are discussed for some locally one-dimensional (LOD) splitting methods applied to linear parabolic initial-boundary value problems. We shall consider unconditional convergence, where both the stepsize in time and the meshwidth in space tend to zero, independently of each other.

**[1]**W. Auzinger,*On error structures and extrapolation for stiff systems, with application in the method of lines*, Computing**44**(1990), no. 4, 331–356 (English, with German summary). MR**1063771**, 10.1007/BF02241272**[2]**Philip Brenner, Michel Crouzeix, and Vidar Thomée,*Single-step methods for inhomogeneous linear differential equations in Banach space*, RAIRO Anal. Numér.**16**(1982), no. 1, 5–26 (English, with French summary). MR**648742****[3]**Miroslav Fiedler,*Special matrices and their applications in numerical mathematics*, Martinus Nijhoff Publishers, Dordrecht, 1986. Translated from the Czech by Petr Přikryl and Karel Segeth. MR**1105955****[4]**P. J. van der Houwen and J. G. Verwer,*One-step splitting methods for semidiscrete parabolic equations*, Computing**22**(1979), no. 4, 291–309 (English, with German summary). MR**620058**, 10.1007/BF02265311**[5]**W. H. Hundsdorfer and B. I. Steininger,*Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems*, BIT**31**(1991), no. 1, 124–143. MR**1097487**, 10.1007/BF01952789**[6]**W. H. Hundsdorfer and J. G. Verwer,*Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems*, Math. Comp.**53**(1989), no. 187, 81–101. MR**969489**, 10.1090/S0025-5718-1989-0969489-7**[7]**Peter Lancaster and Miron Tismenetsky,*The theory of matrices*, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985. MR**792300****[8]**R. J. LeVeque,*Intermediate boundary conditions for LOD, ADI and approximate-factorization methods*, ICASE report 85-21, NASA Langley Research Centre, Hampton, VA, 1985.**[9]**G. I. Marchuk,*Methods of numerical mathematics*, 2nd ed., Applications of Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1982. Translated from the Russian by Arthur A. Brown. MR**661258****[10]**G. I. Marchuk,*Splitting and alternating direction methods*, Handbook of numerical analysis, Vol. I, Handb. Numer. Anal., I, North-Holland, Amsterdam, 1990, pp. 197–462. MR**1039325****[11]**A. A. Samarskiĭ,*An efficient difference method for solving a multidimensional parabolic equation in an arbitrary domain*, Ž. Vyčisl. Mat. i Mat. Fiz.**2**(1962), 787–811 (Russian). MR**0183127****[12]**J. M. Sanz-Serna, J. G. Verwer, and W. H. Hundsdorfer,*Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations*, Numer. Math.**50**(1987), no. 4, 405–418. MR**875165**, 10.1007/BF01396661**[13]**B. P. Sommeijer, P. J. van der Houwen, and J. G. Verwer,*On the treatment of time-dependent boundary conditions in splitting methods for parabolic differential equations*, Internat. J. Numer. Methods Engrg.**17**(1981), no. 3, 335–346. MR**608685**, 10.1002/nme.1620170304**[14]**D. A. Swayne,*Time-dependent boundary and interior forcing in locally one-dimensional schemes*, SIAM J. Sci. Statist. Comput.**8**(1987), no. 5, 755–767. MR**902741**, 10.1137/0908065**[15]**J. G. Verwer,*Contractivity of locally one-dimensional splitting methods*, Numer. Math.**44**(1984), no. 2, 247–259. MR**753957**, 10.1007/BF01410109**[16]**J. G. Verwer and H. B. de Vries,*Global extrapolation of a first order splitting method*, SIAM J. Sci. Statist. Comput.**6**(1985), no. 3, 771–780. MR**791198**, 10.1137/0906052**[17]**N. N. Yanenko,*The method of fractional steps. The solution of problems of mathematical physics in several variables*, Springer-Verlag, New York-Heidelberg, 1971. Translated from the Russian by T. Cheron. English translation edited by M. Holt. MR**0307493**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M12,
65M20

Retrieve articles in all journals with MSC: 65M12, 65M20

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1992-1106972-8

Keywords:
Numerical analysis,
initial-boundary value problems,
LOD methods,
unconditional convergence

Article copyright:
© Copyright 1992
American Mathematical Society