Abstract
In this paper, several splitting methods are discussed which can be used to solve fourth order parabolic partial differential equations that are given in some suitable first order system form. The methods are generalisations of splitting methods for (second order) parabolic PDE's. For all methods which are considered, stability or instability is studied for problems in 2 and in 3 or more spatial dimensions.
Zusammenfassung
In dieser Arbeit werden mehrere Splitting-Methoden diskutiert, welche zur Lösung von parabolischen partiellen Differentialgleichungen vierter Ordnung benützt werden können, welche in Form eines Systems erster Ordnung geschrieben sind. Die Methoden sind Verallgemeinerungen von Splitting-Methoden für parabolische Gleichungen zweiter Ordnung. Für alle betrachteten Methoden werden Stabilitätsuntersuchungen angestellt für den Fall von 2, 3 und mehr Raumdimensionen.
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Fairweather, G., Gourlay, A. R.: Some stable difference approximations to a fourth order parabolic partial differential equation. Maths. of Comp.11, 1–11 (1967).
Gear, C. W.: Numerical initial value problems in ordinary differential equations. Englewood Cliffs, N. J.: Prentice-Hall 1971.
Gourlay, A. R.: Splitting methods for time-dependent partial differential equations. In: The state of the art in numerical analysis (Jacobs, D. A. H., ed.), pp. 757–796. New York: Academic Press 1977.
Gourlay, A. R., Mitchell, A. R.: Intermediate boundary corrections for split operator methods in three dimensions. B. I. T.7, 31–38 (1967).
Gourlay, A. R., Mitchell, A. R.: The equivalence of certain alternating and locally one-dimensional difference methods. SIAM J. Numer. Anal.6, 37–46 (1969).
Gourlay, A. R., Mitchell, A. R.: On the structure of alternating direction implicit (ADI) and locally one dimension (LOD) difference methods. J. I. M. A.9, 80–90 (1972).
van der Houwen, P. J.: Multistep splitting methods of high order for initial value problems. Rep. NW 49/77. Amsterdam: Mathematisch Centrum 1977.
van der Houwen, P. J.: Multistep splitting methods of high order for initial value problems. SIAM J. Numer. Anal.17, 410–417 (1980).
van der Houwen, P. J., Verwer, J. G.: One-step splitting methods for semi-discrete parabolic equations. Computing22, 291–309 (1979).
van der Houwen, P. J., Verwer, J. G.: Comparison of algorithms for systems of ordinary differential equations originating from parabolic initial boundary value problems in two space dimensions. In: Performance evaluation of numerical software (Fosdick, L. D., ed.), pp. 185–198. Amsterdam: North-Holland Publishing Comp. 1979.
Janenko, N. N.: Die Zwischenschrittmethode zur Lösung mehrdimensionaler Probleme der mathematischen Physik. Lect. Notes in Maths. 91. Berlin: Springer-Verlag 1969.
Jeltsch, R.: Stability on the imaginary axis andA-stability of linear multistep methods. B. I. T.18, 170–174 (1978).
Joubert, G. R.: Explicit difference methods for the solution of the equation of a vibrating rod. In: Numerische Behandlung von Differentialgleichungen II, Tagung Oberwolfach 17–22. 11. 1975, ISNM 31. Birkhäuser-Verlag 1976, pp. 91–104.
Kellogg, R. B.: An alternating direction method for operator equations. J. SIAM12, 848–854 (1964).
Marchuk, G. I.: On the theory of the splitting-up method. In: Numerical solutions of partial differential equations II, Synspade, 1970 (Hubbard, B., ed.), pp. 469–500. New York: Academic Press 1971.
ter Maten, E. J. W.: Stability analysis of finite difference methods for fourth order parabolic partial differential equations. Academisch Proefschrift, Univ. of Utrecht 1984.
ter Maten, E. J. W., Sleijpen, G. L. G.: Hopscotch methods for fourth order parabolic equations I: stability results for fixed stepsizes. Preprint 275, Math. Inst., Univ. of Utrecht 1983.
ter Maten, E. J. W., Sleijpen, G. L. G.: A convergence analysis of hopscotch methods for fourth order parabolic equations. To appear in Numer. Math.
McGuire, G. R., Morris, J. L.: Restoring orders of accuracy for multi-level schemes for non-linear hyperbolic systems in many space variables. J. I. M. A.17, 53–67 (1976).
Miller, J. J. H.: On the location of zero's of certain classes of polynomials with applications to numerical analysis. J. I. M. A.8, 397–406 (1971).
Mitchell, A. R.: Computational methods in partial differential equations. London: J. Wiley & Sons 1969.
Morris, J. L.: Splitting methods for parabolic and hyperbolic partial differential equations. In: Numerische Methoden bei Differentialgleichungen und mit funktionalanalytischen Hilfsmitteln, Oberwolfach 31. 5.–2. 6. 1972, ISNM 19, pp. 169–180. Basel: Birkhäuser-Verlag 1974.
van der Sluis, A.: Numerieke analyse III (Numerieke behandeling van differentiaalvergelijkingen) (in dutch). College diktaat, Math. Inst., Univ. of Utrecht 1979.
Sommeyer, B. P., van der Houwen, P. J., Verwer, J. G.: On the treatment of time-dependent boundary conditions in splitting methods for parabolic differential equations. Intern. J. Numer. Math. Engin.17, 335–346 (1981).
Warming, R. F., Beam, R. M.: An extension to alternating direction implicit methods. B. I. T.19, 395–417 (1979).
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ter Maten, E.J.W. Splitting methods for fourth order parabolic partial differential equations. Computing 37, 335–350 (1986). https://doi.org/10.1007/BF02251091
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DOI: https://doi.org/10.1007/BF02251091