An analysis of operator splitting techniques in the stiff case.

*(English)*Zbl 0953.65062Operator splitting methods are widely used in many applications, such as air pollution modeling, combustion, and reactive flows. The author regards the case where the evolution equations to be simulated are stiff. He considers systems with two operators: a stiff one and a non-stiff one. For example, a linear evolution system under a singular perturbation has the form
\[
{{d z}\over{d t}} = \Biggl({A\over \varepsilon} + B \Biggr) z,\quad z(0) = z_0,
\]
where \(\varepsilon\) is a small positive parameter. The author shows some splitting schemes for solving of the evolution systems (“first-order”, “second-order” schemes). The “first-order” scheme has the form
\[
\begin{aligned} {{d z^*}\over{d t}} = B^* z^*,\quad z^*(0) = z_0, &\quad \text{on }[0,\Delta t],\\ {{d z^{**}}\over{d t}} = B^* z^{**},\quad z^{**}(0) = z^*(\Delta t), &\quad \text{on }[0,\Delta t],\end{aligned}
\]
where the final value is given by \(z^{**}(\Delta t)\). The main results deals with the choice of the sequential order for the operators: the stiff operator must always be last in the splitting scheme.

Reviewer: A.Dishliev (Sofia)

##### MSC:

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

92D40 | Ecology |

92E20 | Classical flows, reactions, etc. in chemistry |

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

34E15 | Singular perturbations, general theory for ordinary differential equations |

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

65L05 | Numerical methods for initial value problems |

80A25 | Combustion |

80M20 | Finite difference methods applied to problems in thermodynamics and heat transfer |

##### Keywords:

operator splitting; Air Pollution Modeling; combustion; reactive flows; stiff systems; semidiscretization; linear evolution system; singular perturbation##### Software:

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\textit{B. Sportisse}, J. Comput. Phys. 161, No. 1, 140--168 (2000; Zbl 0953.65062)

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