A note on asymptotic splitting and its applications.

*(English)*Zbl 0822.65046The authors actually consider the initial value problem for the system of ordinary differential equations \(v_ t= \sum^ n_{i= 1} A_ i v\), \(0< t\leq T\), with constant matrices \(A_ i\), \(i= 1,\dots, n\).

The standard representation \(v(t)= Pv(0)\) with \(P\equiv P(t)= \exp\left\{t \sum^ n_{i= 1} A_ i\right\}\) suggests various approximations. Special attention is paid to the so-called “generalized asymptotic splitting” (it might yield better accuracy in a sense).

It should be noted that, e.g., the case of approximations for parabolic problems needs taking into account the fact that the above matrices depend strongly on \(h\) (on the parameter of the space grid) and that approximations of the boundary conditions are crucial.

The standard representation \(v(t)= Pv(0)\) with \(P\equiv P(t)= \exp\left\{t \sum^ n_{i= 1} A_ i\right\}\) suggests various approximations. Special attention is paid to the so-called “generalized asymptotic splitting” (it might yield better accuracy in a sense).

It should be noted that, e.g., the case of approximations for parabolic problems needs taking into account the fact that the above matrices depend strongly on \(h\) (on the parameter of the space grid) and that approximations of the boundary conditions are crucial.

Reviewer: E.D’yakonov (Moskva)

##### MSC:

65L05 | Numerical methods for initial value problems |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

34A30 | Linear ordinary differential equations and systems, general |

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\textit{Q. Sheng} and \textit{R. P. Agarwal}, Math. Comput. Modelling 20, No. 12, 45--58 (1994; Zbl 0822.65046)

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