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**Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations.**
*(English)*
Zbl 1144.65062

Computational simulations of bioremediation, heat transfer, transport and chemical reaction problems require accurate numerical methods. One of the widely spread methods is to transform time and space dependent problems described by one or a system of partial differential equations to a system of ordinary differential equations (ODEs) using a certain space discretization method like the finite element method and then solve the system of ODEs.

The presented method is a combination of two efficient methods: an iterative and an operator-splitting method by using the advantages of both of them. Splitting methods have the benefit in decoupling multiphysics problems into simpler physical problems and solve this reduced problems in each time step. The iterative method is used in each time step for solving reduced problems iteratively. An analysis of the convergence and the rate of convergence of this iterative operator-splitting method is presented.

To obtain high-order methods Runge-Kutta and backward differentiation formula methods are used. For both commutative and noncommutative operator of the continuous equations the A-stability of the method is proved under certain conditions. Various numerical experiments for systems of ODEs or parabolic equations also with nonlinearities are presented. Comparisons of various methods of higher order together with various splitting methods are shown.

The presented method is a combination of two efficient methods: an iterative and an operator-splitting method by using the advantages of both of them. Splitting methods have the benefit in decoupling multiphysics problems into simpler physical problems and solve this reduced problems in each time step. The iterative method is used in each time step for solving reduced problems iteratively. An analysis of the convergence and the rate of convergence of this iterative operator-splitting method is presented.

To obtain high-order methods Runge-Kutta and backward differentiation formula methods are used. For both commutative and noncommutative operator of the continuous equations the A-stability of the method is proved under certain conditions. Various numerical experiments for systems of ODEs or parabolic equations also with nonlinearities are presented. Comparisons of various methods of higher order together with various splitting methods are shown.

Reviewer: Angela Handlovičová (Bratislava)

### MSC:

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

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

### Keywords:

operator-splitting methods; iterative methods; explicit and implicit time discretization methods; stability; consistency; stiff differential equations; Runge-Kutta method; convergence; numerical experiments; parabolic equations
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\textit{J. Geiser}, J. Comput. Appl. Math. 217, No. 1, 227--242 (2008; Zbl 1144.65062)

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