# On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six

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## Abstract

**:**

## 1. Introduction

- It contains widely varying time scales, i.e., some components of the solution decay much more rapidly than others [3];
- The step size is dictated by the stability requirements rather than the accuracy requirements [3];
- Explicit methods do not work, or work only extremely slowly [4];
- All of its eigenvalues have negative real parts, and the stiffness ratio (the ratio of the magnitudes of the real parts of the largest and smallest eigenvalues) is large [3];
- No solution component is unstable (no eigenvalue of the Jacobian matrix has a real part which is at all large and positive) and at least some components are very stable (at least one eigenvalue has a real part which is large and negative) [5].

- (i)
- $Re\left({\lambda}_{i}\right)<0,\text{}i=1,2,\dots ,d$ and
- (ii)
- $\underset{i}{\mathrm{max}}\left|Re\left({\lambda}_{i}\right)\right|\gg \underset{i}{\mathrm{min}}\left|Re\left({\lambda}_{i}\right)\right|,$ where ${\lambda}_{i}$ are the eigenvalues of A and the ratio $S=\underset{i}{\mathrm{max}}\left|Re\left({\lambda}_{i}\right)\right|/\underset{i}{\mathrm{min}}\left|Re\left({\lambda}_{i}\right)\right|$ is called the stiffness ratio as a measure of stiffness.

## 2. Derivation of BBDF with Off-Step Points, BBDFO

## 3. Order, Convergence and Stability of the Method

**Definition**

**3.1.**

**Definition**

**3.2.**

**Definition**

**3.3.**

**Definition**

**3.4.**

**Definition**

**3.5.**

#### 3.1. Order of the Method

#### 3.2. Zero Stability of the Method

#### 3.3. Stability Region of the Method

#### 3.4. Stability Comparison

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BBDFO(6) | New Block Backward Differentiation Formula with off-step points of order 6 |

BBDF(6) | 2-point Block Backward Differentiation Formula of order 6 in [28] |

ode15s | Variable order Backward Differentiation Formula [7]. |

NS | Number of steps |

$h$ | Step size |

MAXE | Maximum error |

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**Figure 2.**Stability region of the sixth order Block Backward Differentiation Formula with two off-step points (BBDFO(6)).

**Figure 3.**Comparison of the stability regions between the BBDFO(6) and the Block Backward Differentiation Formula without the off-step points (BBDF(6)).

Method | Interval of Instability |
---|---|

BBDFO(6) | (0, 10.05) |

BBDF(6) | (0, 17.10) |

$\mathit{h}.$ | Methods | NS | MAXE |
---|---|---|---|

${10}^{-3}$ | BBDFO(6) | 2.11157($-$2) | |

BBDF(6) | 9.11882($-$4) | ||

ode15s | 2.08844($-$3) | ||

${10}^{-4}$ | BBDFO(6) | 5.54678($-$3) | |

BBDF(6) | 1.69293($-$2) | ||

ode15s | 2.60950($-$4) | ||

${10}^{-5}$ | BBDFO(6) | 7.38966($-$5) | |

BBDF(6) | 3.02723($-$4) | ||

ode15s | 3.76862($-$5) | ||

${10}^{-6}$ | BBDFO(6) | 7.60256($-$7) | |

BBDF(6) | 3.20426($-$6) | ||

ode15s | 6.32160($-$6) |

$\mathit{h}.$ | Methods | NS | MAXE |
---|---|---|---|

${10}^{-3}$ | BBDFO(6) | 5.68483($-$7) | |

BBDF(6) | 2.38845($-$6) | ||

ode15s | 9.37878($-$4) | ||

${10}^{-4}$ | BBDFO(6) | 5.71640($-$9) | |

BBDF(6) | 2.41537($-$8) | ||

ode15s | 1.14126($-$4) | ||

${10}^{-5}$ | BBDFO(6) | 5.71960($-$11) | |

BBDF(6) | 2.41808($-$10) | ||

ode15s | 1.75037($-$5) | ||

${10}^{-6}$ | BBDFO(6) | 9.52614($-$11) | |

BBDF(6) | 7.40070($-$11) | ||

ode15s | 2.61573($-$6) |

$\mathit{h}.$ | Methods | NS | MAXE |
---|---|---|---|

${10}^{-3}$ | BBDFO(6) | 2.04408($-$3) | |

BBDF(6) | 7.62867($-$3) | ||

ode15s | 8.69860($-$4) | ||

${10}^{-4}$ | BBDFO(6) | 2.28504($-$5) | |

BBDF(6) | 9.54280($-$5) | ||

ode15s | 1.21447($-$4) | ||

${10}^{-5}$ | BBDFO(6) | 2.31054($-$7) | |

BBDF(6) | 9.75695($-$7) | ||

ode15s | 1.36101($-$5) | ||

${10}^{-6}$ | BBDFO(6) | 2.31311($-$9) | |

BBDF(6) | 9.77861($-$9) | ||

ode15s | 2.85643($-$6) |

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**MDPI and ACS Style**

Nasarudin, A.A.; Ibrahim, Z.B.; Rosali, H.
On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six. *Symmetry* **2020**, *12*, 952.
https://doi.org/10.3390/sym12060952

**AMA Style**

Nasarudin AA, Ibrahim ZB, Rosali H.
On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six. *Symmetry*. 2020; 12(6):952.
https://doi.org/10.3390/sym12060952

**Chicago/Turabian Style**

Nasarudin, Amiratul Ashikin, Zarina Bibi Ibrahim, and Haliza Rosali.
2020. "On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six" *Symmetry* 12, no. 6: 952.
https://doi.org/10.3390/sym12060952