Stability results for fractional step discretizations of time dependent coefficient evolutionary problems

Abstract We consider a class of additive Runge–Kutta methods, which include most of the classical alternating direction or fractionary step methods, for discretizing the time variable in an evolutionary problem whose coefficients depend on time. Some stability results are proven for these methods which, together with suitable consistency properties, permit us to show the convergence of these discretizations.

[1]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[2]  P. J. van der Houwen,et al.  The development of Runge-Kutta methods for partial differential equations , 1996 .

[3]  C. Palencia,et al.  A stability result for sectorial operators in branch spaces , 1993 .

[4]  A. Sayfy,et al.  Additive methods for the numerical solution of ordinary differential equations , 1980 .

[5]  Unitary Dilations of Hilbert Space Operators and Related Topics , 1974 .

[6]  J. C. Jorge,et al.  Contractivity results for alternating direction schemes in Hilbert spaces , 1994 .

[7]  V. Thomée,et al.  ON RATIONAL APPROXIMATIONS OF SEMIGROUPS , 1979 .

[8]  G. J. Cooper,et al.  Additive Runge-Kutta methods for stiff ordinary differential equations , 1983 .

[9]  Blanca Bujanda Cirauqui.  Métodos Runge-Kutta de pasos fraccionarios de orden alto para la resolución de problemas evolutivos de corrección-difusión-reacción , 1999 .

[10]  Juan Carlos Jorge Ulecia.  Los métodos de pasos fraccionarios para la integración de problemas parabólicos lineales. Formulación general análisis de la convergencia y diseño de nuevos métodos , 1992 .

[11]  Willem Hundsdorfer,et al.  Trapezoidal and midpoint splittings for initial-boundary value problems , 1998, Math. Comput..

[12]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[13]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[14]  Stig Larsson,et al.  The stability of rational approximations of analytic semigroups , 1993 .

[15]  Cesar Palencia,et al.  Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case , 1999, Math. Comput..