### Abstract

Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods. The computational cost per step of these new numerical methods is higher than the computational cost per step of the underlying ERKMs. However, the order of accuracy of the combined methods becomes very high: if the order of accuracy of the underlying ERKM is p, then the order of accuracy of its combination with the Two-times Repeated Richardson Extrapolation is at least p+3 when the right-hand-side function of the system of ODEs is sufficiently many times continuously differentiable. Moreover, the stability properties of the new methods are always better than those of the underlying numerical methods when p=m and m=1,2,3,4 (where m is the number of stage vectors in the chosen ERKM). These two useful properties, higher accuracy and better stability, are often giving a very reasonable compensation for the increased computational cost per step, because the same degree of accuracy can be achieved by applying a large stepsize which leads to a considerable reduction of the number of steps when the Two-times Repeated Richardson Extrapolation is used. This fact is verified by several numerical experiments.

Original language | English |
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Title of host publication | Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers |

Editors | István Faragó, Ivan Dimov, Lubin Vulkov |

Publisher | Springer Verlag |

Pages | 678-686 |

Number of pages | 9 |

ISBN (Print) | 9783030115388 |

DOIs | |

Publication status | Published - Jan 1 2019 |

Event | 7th International Conference on Finite Difference Methods, FDM 2018 - Lozenetz, Bulgaria Duration: Jun 11 2018 → Jun 16 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11386 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 7th International Conference on Finite Difference Methods, FDM 2018 |
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Country | Bulgaria |

City | Lozenetz |

Period | 6/11/18 → 6/16/18 |

### Fingerprint

### Keywords

- Absolute stability properties
- Explicit Runge-Kutta Methods
- Systems of ordinary differential equations (ODEs)
- Two-times Repeated Richardson Extrapolation

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers*(pp. 678-686). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11386 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-11539-5_80

**Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit Runge-Kutta Methods.** / Zlatev, Zahari; Dimov, Ivan; Faragó, I.; Georgiev, Krassimir; Havasi, Ágnes.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11386 LNCS, Springer Verlag, pp. 678-686, 7th International Conference on Finite Difference Methods, FDM 2018, Lozenetz, Bulgaria, 6/11/18. https://doi.org/10.1007/978-3-030-11539-5_80

}

TY - GEN

T1 - Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit Runge-Kutta Methods

AU - Zlatev, Zahari

AU - Dimov, Ivan

AU - Faragó, I.

AU - Georgiev, Krassimir

AU - Havasi, Ágnes

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods. The computational cost per step of these new numerical methods is higher than the computational cost per step of the underlying ERKMs. However, the order of accuracy of the combined methods becomes very high: if the order of accuracy of the underlying ERKM is p, then the order of accuracy of its combination with the Two-times Repeated Richardson Extrapolation is at least p+3 when the right-hand-side function of the system of ODEs is sufficiently many times continuously differentiable. Moreover, the stability properties of the new methods are always better than those of the underlying numerical methods when p=m and m=1,2,3,4 (where m is the number of stage vectors in the chosen ERKM). These two useful properties, higher accuracy and better stability, are often giving a very reasonable compensation for the increased computational cost per step, because the same degree of accuracy can be achieved by applying a large stepsize which leads to a considerable reduction of the number of steps when the Two-times Repeated Richardson Extrapolation is used. This fact is verified by several numerical experiments.

AB - Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods. The computational cost per step of these new numerical methods is higher than the computational cost per step of the underlying ERKMs. However, the order of accuracy of the combined methods becomes very high: if the order of accuracy of the underlying ERKM is p, then the order of accuracy of its combination with the Two-times Repeated Richardson Extrapolation is at least p+3 when the right-hand-side function of the system of ODEs is sufficiently many times continuously differentiable. Moreover, the stability properties of the new methods are always better than those of the underlying numerical methods when p=m and m=1,2,3,4 (where m is the number of stage vectors in the chosen ERKM). These two useful properties, higher accuracy and better stability, are often giving a very reasonable compensation for the increased computational cost per step, because the same degree of accuracy can be achieved by applying a large stepsize which leads to a considerable reduction of the number of steps when the Two-times Repeated Richardson Extrapolation is used. This fact is verified by several numerical experiments.

KW - Absolute stability properties

KW - Explicit Runge-Kutta Methods

KW - Systems of ordinary differential equations (ODEs)

KW - Two-times Repeated Richardson Extrapolation

UR - http://www.scopus.com/inward/record.url?scp=85066129673&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066129673&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-11539-5_80

DO - 10.1007/978-3-030-11539-5_80

M3 - Conference contribution

SN - 9783030115388

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 678

EP - 686

BT - Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers

A2 - Faragó, István

A2 - Dimov, Ivan

A2 - Vulkov, Lubin

PB - Springer Verlag

ER -