Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case

González, C.; Palencia, C.

doi:10.1090/S0025-5718-99-01018-2

Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case
HTML articles powered by AMS MathViewer

by C. González and C. Palencia PDF

Math. Comp. 68 (1999), 73-89 Request permission

Abstract:

We consider an abstract time-dependent, linear parabolic problem \[ u’(t) = A(t)u(t), \qquad u(t_0) = u_0, \] where $A(t) : D \subset X \to X$, $t \in J$, is a family of sectorial operators in a Banach space $X$ with time-independent domain $D$. This problem is discretized in time by means of an A($\theta$) strongly stable Runge-Kutta method, $0 < \theta <\pi /2$. We prove that the resulting discretization is stable, under the assumption \[ \| (A(t) - A(s) )x \| \le L|t-s|^\alpha (\|x\|+ \| A(s)x\|), \qquad x\in D, t, s \in J, \] where $L>0$ and $\alpha \in (0,1)$. Our results are applicable to the analysis of parabolic problems in the $L^p$, $p \ne 2$, norms.

References

Paolo Acquistapace, Abstract linear nonautonomous parabolic equations: a survey, Differential equations in Banach spaces (Bologna, 1991) Lecture Notes in Pure and Appl. Math., vol. 148, Dekker, New York, 1993, pp. 1–19. MR 1236683
Herbert Amann, Parabolic evolution equations in interpolation and extrapolation spaces, J. Funct. Anal. 78 (1988), no. 2, 233–270. MR 943499, DOI 10.1016/0022-1236(88)90120-6
Herbert Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations 72 (1988), no. 2, 201–269. MR 932367, DOI 10.1016/0022-0396(88)90156-8
N. Yu. Bakaev, Some problems on the correctness of difference schemes on nonuniform grids, Zh. Vychisl. Mat. i Mat. Fiz. 33 (1993), no. 4, 561–577 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 33 (1993), no. 4, 511–524. MR 1217954
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
Philip Brenner and Vidar Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), no. 4, 683–694. MR 537280, DOI 10.1137/0716051
M. Crouzeix, S. Larsson, S. Piskarëv, and V. Thomée, The stability of rational approximations of analytic semigroups, BIT 33 (1993), no. 1, 74–84. MR 1326004, DOI 10.1007/BF01990345
Hector O. Fattorini, The Cauchy problem, Encyclopedia of Mathematics and its Applications, vol. 18, Addison-Wesley Publishing Co., Reading, Mass., 1983. With a foreword by Felix E. Browder. MR 692768
H. Fujita and T. Suzuki, Evolution Problems, Handbook of numerical analysis, Ed. P. G. Ciarlet and J. L. Lions, North-Holland, Amsterdam, 1991.
C. González and C. Palencia, Stability of time-stepping methods for time-dependent parabolic problems, SIAM J. Numer. Anal., 35 (1998), pp. 973–989.
C. González and C. Palencia, Stability of Runge–Kutta methods for abstract quasilinear parabolic problems in Banach spaces, submitted to Math. Comp.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff Problems, Springer-Verlag, Berlin (1988).
S. G. Kreĭn, Linear differential equations in Banach space, Translations of Mathematical Monographs, Vol. 29, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin. MR 0342804
S. Larsson, V. Thomée and L. B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Preprint 1995:5, Department of Mathematics, Chalmers University of Technology and Göteborg University, Göteborg, 1995.
Christian Lubich and Alexander Ostermann, Runge-Kutta approximation of quasi-linear parabolic equations, Math. Comp. 64 (1995), no. 210, 601–627. MR 1284670, DOI 10.1090/S0025-5718-1995-1284670-0
C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal. 30 (1993), no. 5, 1373–1384. MR 1239826, DOI 10.1137/0730071
C. Palencia, Stability of rational multistep approximations of holomorphic semigroups, Math. Comp. 64 (1995), no. 210, 591–599. MR 1277770, DOI 10.1090/S0025-5718-1995-1277770-2
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
P. E. Sobolevskii, Equations of parabolic type in Banach space, Amer. Math. Soc. Transl., 49 (1966), pp. 1-62.
P. E. Sobolevskii, The theory of semigroups and stability of difference schemes, Culc. c. of Sibirien AN USSR. School on theory of operators in functional spaces (25-31 August), Novosibirsk, Preprint, (1975), pp. 1-38.
I. A. Aliev, Riesz transforms generated by a generalized translation operator, Izv. Akad. Nauk Azerbaĭdzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk 8 (1987), no. 1, 7–13 (Russian, with English and Azerbaijani summaries). MR 939101
H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580