Stability of Runge-Kutta methods for quasilinear parabolic problems

González, C.; Palencia, C.

doi:10.1090/S0025-5718-99-01156-4

Stability of Runge-Kutta methods for quasilinear parabolic problems
HTML articles powered by AMS MathViewer

by C. González and C. Palencia PDF

Math. Comp. 69 (2000), 609-628 Request permission

Abstract:

We consider a quasilinear parabolic problem \[ u’(t) = Q\big ( u(t) \big ) u(t), \qquad u(t_0) = u_0 \in \mathcal {D}, \] where $Q(w) : \mathcal {D}\subset X \to X$, $w \in W \subset X$, is a family of sectorial operators in a Banach space $X$ with fixed domain $\mathcal {D}$. This problem is discretized in time by means of a strongly A($\theta$)-stable, $0 < \theta \le \pi /2$, Runge–Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the dependence of $Q(w)$ with respect to $w$. Our results are useful for studying in $L^p$ norms, $1 \le p \le + \infty$, many problems arising in applications. Some auxiliary results for time-dependent parabolic problems are also provided.

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
H. A. Alibekov and P. E. Sobolevskiĭ, Stability and convergence of high-order difference schemes of approximation for parabolic equations, Ukrain. Mat. Zh. 31 (1979), no. 6, 627–634, 765 (Russian). MR 567277
Herbert Amann, Quasilinear evolution equations and parabolic systems, Trans. Amer. Math. Soc. 293 (1986), no. 1, 191–227. MR 814920, DOI 10.1090/S0002-9947-1986-0814920-4
Herbert Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal. 12 (1988), no. 9, 895–919. MR 960634, DOI 10.1016/0362-546X(88)90073-9
Herbert Amann, Highly degenerate quasilinear parabolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 1, 135–166. MR 1118224
Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385, DOI 10.1007/978-3-0348-9221-6
N. Bakaev, On the stability of nonlinear difference problems constructed by the Runge–Kutta method, deposited in VINITI, No. 1507-V91, 1991.
Jonathan Bell, Avner Friedman, and Andrew A. Lacey, On solutions to a quasi-linear diffusion problem from the study of soft tissue, SIAM J. Appl. Math. 51 (1991), no. 2, 484–493. MR 1095030, DOI 10.1137/0151024
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
Ph. Clément, C. J. van Duijn, and Shuanhu Li, On a nonlinear elliptic-parabolic partial differential equation system in a two-dimensional groundwater flow problem, SIAM J. Math. Anal. 23 (1992), no. 4, 836–851. MR 1166560, DOI 10.1137/0523044
Donald S. Cohen and Andrew B. White Jr., Sharp fronts due to diffusion and viscoelastic relaxation in polymers, SIAM J. Appl. Math. 51 (1991), no. 2, 472–483. MR 1095029, DOI 10.1137/0151023
M. Crouzeix, S. Larsson, and V. Thomée, Resolvent estimates for elliptic finite element operators in one dimension, Math. Comp. 63 (1994), no. 207, 121–140. MR 1242058, DOI 10.1090/S0025-5718-1994-1242058-1
C. González and C. Palencia, Stability of time-stepping methods for abstract time-dependent parabolic problems, SIAM J. Numer. Anal. 35 (1998), no. 3, 973–989. MR 1619918, DOI 10.1137/S0036142995283412
C. González and C. Palencia, Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: the Hölder case, Math. Comp. 68 (1999), no. 225, 73–89. MR 1609666, DOI 10.1090/S0025-5718-99-01018-2
E. Hairer and G. Wanner, Solving ordinary differential equations. II, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1996. Stiff and differential-algebraic problems. MR 1439506, DOI 10.1007/978-3-642-05221-7
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
Christian Lubich and Alexander Ostermann, Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour, Appl. Numer. Math. 22 (1996), no. 1-3, 279–292. Special issue celebrating the centenary of Runge-Kutta methods. MR 1424303, DOI 10.1016/S0168-9274(96)00038-4
Alessandra Lunardi, Global solutions of abstract quasilinear parabolic equations, J. Differential Equations 58 (1985), no. 2, 228–242. MR 794770, DOI 10.1016/0022-0396(85)90014-2
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
A. Ostermann and M. Roche, Runge-Kutta methods for partial differential equations and fractional orders of convergence, Math. Comp. 59 (1992), no. 200, 403–420. MR 1142285, DOI 10.1090/S0025-5718-1992-1142285-6
C. Palencia, Maximum norm analysis of completely discrete finite element methods for parabolic problems, SIAM J. Numer. Anal. 33 (1996), no. 4, 1654–1668. MR 1403564, DOI 10.1137/S0036142993259779
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. Sobolevskiĭ, Equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obšč. 10 (1961), 297–350 (Russian). MR 0141900
Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. MR 533824
Dunham Jackson, A class of orthogonal functions on plane curves, Ann. of Math. (2) 40 (1939), 521–532. MR 80, DOI 10.2307/1968936