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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 $$ u'(t)=Au(t)+f(t,u(t)),\quad u(0)=u_0,\quad 0\leq t\leq T.$$ 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.

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
Full Text: DOI
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