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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

${u}^{\text{'}}\left(t\right)=Au\left(t\right)+f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)={u}_{0},\phantom{\rule{1.em}{0ex}}0\le t\le T·$

The authors assume that $A:D\left(A\right)\subset X\to X$ is an infinitesimal generator of a ${C}_{0}$-semigroup ${e}^{tA},\phantom{\rule{4pt}{0ex}}t\ge 0$, of linear and bounded operators in a complex Banach space $X$, with growth governed by $\parallel {e}^{tA}\parallel \le M{e}^{\omega t},\phantom{\rule{4pt}{0ex}}t\ge 0$ for some $M>0,\omega \in ℝ$. 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.

##### MSC:
 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
Octave
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