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Zbl 1087.65054
Calvo, M.P.; Palencia, C.
A class of explicit multistep exponential integrators for semilinear problems. (English)
[J] Numer. Math. 102, No. 3, 367-381 (2006). ISSN 0029-599X; ISSN 0945-3245/e

The authors derive and analyze a family of explicit multistep exponential methods for the time integration of abstract semilinear problems $$ u'(t)=Au(t)+f(t,u(t)),\quad u(0)=u_0,\quad 0\leq t\leq T.$$ The authors assume that $A:D(A)\subset X \to X$ is an infinitesimal generator of a $C_0$-semigroup $e^{tA},\ t\geq 0$, of linear and bounded operators in a complex Banach space $X$, with growth governed by $\Vert e^{tA}\Vert\leq Me^{\omega t},\ t\geq 0$ for some $M>0,\omega \in \Bbb R$. 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.
[Karel Najzar (Praha)]

MSC 2000:: *65J15 Equations with nonlinear operators (numerical methods)
65L05 Initial value problems for ODE (numerical methods)
65L06 Multistep, Runge-Kutta, and extrapolation methods
65M12 Stability and convergence of numerical methods (IVP of PDE)
65M20 Method of lines (IVP of PDE)
34G20 Nonlinear ODE in abstract spaces

Keywords: Banach space; abstract semilinear problem; explicit multistep exponential methods; numerical examples

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