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A class of explicit multistep exponential integrators for semilinear problems. (English) Zbl 1087.65054

The authors derive and analyze a family of explicit multistep exponential methods for the time integration of abstract semilinear problems

u ' (t)=Au(t)+f(t,u(t)),u(0)=u 0 ,0tT·

The authors assume that A:D(A)XX is an infinitesimal generator of a C 0 -semigroup e tA ,t0, of linear and bounded operators in a complex Banach space X, with growth governed by e tA Me ωt ,t0 for some M>0,ω. It is shown that the k-step method achieves order k, for appropriate starting values, which can be computed by auxiliary routines or by one strategy proposed in the paper. Together with some implementation issues, numerical illustrations are also provided.

MSC:
65J15Equations with nonlinear operators (numerical methods)
65L05Initial value problems for ODE (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods
65M12Stability and convergence of numerical methods (IVP of PDE)
65M20Method of lines (IVP of PDE)
34G20Nonlinear ODE in abstract spaces
Software:
Octave
References:
[1]Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147, 362–387 (1998)
[2]Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)
[3]Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)
[4]Friesner, R.A., Tuckerman, L.S., Dornblaser, B.C., Russo, T.V.: A method of exponential propagation of large systems of stiff nonlinear differential equations. J. Sci. Comp. 4, 327–354 (1989)
[5]González, C., Palencia, C.: Stability of Runge-Kutta methods for quasilinear parabolic problems. Math. Comput. 69, 609–628 (2000)
[6]Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840, Springer, Berlin, 1981
[7]Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical Society, Providence, 1957
[8]Hochbruck, M., Lubich, Ch.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)
[9]Hochbruck, M., Lubich, Ch.: Error analysis of Krylov methods in a nutshell. SIAM J. Sci. Comput. 19, 695–701 (1998)
[10]Hochbruck, M., Lubich, Ch.: Exponential integrators for quantum-classical molecular dynamics. BIT 39, 620–645 (1999)
[11]Hochbruck, M., Lubich, Ch., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. 19, 1552–1574 (1998)
[12]Hochbruck, M., Ostermann, A.: Exponential Runge-Kutta methods for parabolic problems. Appl. Numer. Math. (to appear)
[13]Hochbruck, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. Preprint, 2004
[14]Kassam, A.-K., Trefethen, L.: Fourth-order time stepping for stiff PDEs. SIAM J. Sci. Comput. (to appear)
[15]Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 72–88 (2005)
[16]Lawson, J.D.: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–380 (1967)
[17]Le Roux, M.N.: Méthodes multipas pour des équations paraboliques non linéaires. Numer. Math. 35, 143–162 (1980)
[18]Lubich, Ch., Ostermann, A.: Runge-Kutta approximation of quasilinear parabolic equations. Math. Comput. 64, 601–627 (1995)
[19]Lubich, Ch., Ostermann, A.: Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behavior. Appl. Numer. Math. 22, 179–292 (1996)
[20]McKee, S.: Generalised discrete Gronwall lemmas. Z. Angew. Math. Mech. 62, 429–434 (1982)
[21]Minchev, B.V., Wright, W.M.: A review of exponential integrators. Preprint, 2004
[22]Nørsett, S.P.: An A-stable modification of the Adams-Bashforth methods. Springer Lect. Notes Math. 109, 214–219 (1969)
[23]Ostermann, A., Thalhammer, M.: Non-smooth data error estimates for linearly implicit Runge-Kutta methods. IMA J. Numer. Anal. 20, 167–184 (2000)
[24]Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983
[25]Quarteroni, A., Saleri, F.: Scientific Computing with MATLAB. Springer, Berlin, 2003
[26]Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978
[27]Van der Houwen, P.J., Verwer, J.G.: Generalized linear multistep methods I. Development of algorithms with zero-parasitic roots. Report NW 10/74, Mathematisch Centrum, Amsterdam, 1974
[28]Verwer, J.G.: Generalized linear multistep methods II. Numerical applications. Report NW 12/74, Mathematisch Centrum, Amsterdam, 1974
[29]Verwer, J.G.: On generalized linear multistep methods with zero-parasitic roots and an adaptive principal root. Numer. Math. 27, 143–155 (1977)