# Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- $Re\left({\lambda}_{i}\right)<0$ and
- (ii)
- $\underset{i}{max}|Re\left({\lambda}_{i}\right)|\gg \underset{i}{min}\left|Re\left({\lambda}_{i}\right)\right|$ where the ratio $\frac{\underset{i}{max}|Re\left({\lambda}_{i}\right)}{\underset{i}{min}\left|Re\left({\lambda}_{i}\right)\right|}$ is called the stiffness ratio or stiffness index.

## 2. The $\mathbf{\rho}$-Diagonally Implicit Block Backward Differentiation Formula

## 3. Selection of Parameter $\mathbf{\rho}$

**Lemma**

**1.**

- (i)
- ${a}_{i}\phantom{\rule{3.33333pt}{0ex}}areeitherallpositiveornegative$,
- (ii)
- ${a}_{1}{a}_{2}-{a}_{0}{a}_{3}>0.$

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

- (i)
- The method is ${A}_{0}$-stable,
- (ii)
- The modulus of any root of the polynomial $\frac{\varrho \left(\xi \right)}{\xi -1}$ is less than 1,
- (iii)
- The roots of $\sigma \left(\xi \right)$ of modulus 1 are simple,
- (iv)
- If ${\xi}_{0}$ is a root of $\sigma \left(\xi \right)$ with $|{\xi}_{0}|=1$, then $\frac{\varrho \left(\xi \right)}{\xi {\sigma}^{\prime}\left(\xi \right)}$ at $\xi ={\xi}_{0}$ is real and positive.

**Lemma**

**2.**

- (i)
- |a
_{0}| < |a_{2}| , - (ii)
- |a
_{1}| < |a_{2}+ a_{0}| .

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 3.1. Zero Stability

**Definition**

**1.**

#### 3.2. Absolute Stability

**Definition**

**2.**

**Definition**

**3.**

#### 3.3. Order of the Method and Error Constant

**Definition**

**4.**

#### 3.4. Convergence

- (a)
- ${\sum}_{j=0}^{k+2}{\alpha}_{j-2,k}=0,$
- (b)
- ${\sum}_{j=0}^{k+2}j{\alpha}_{j-2,k}={\sum}_{j=0}^{k+2}{\beta}_{j,k}.$

## 4. Implementation of the Method

## 5. Numerical Results

**Test Problem 1:**

**Test Problem 2:**

**Test Problem 3:**

**Test Problem 4:**

H : | Step size |

MAXE : | Maximum error |

TIME : | Execution time (seconds) |

ρ-DIBBDF(ρ_{i}) : | ρ-Diagonally Implicit Block Backward Differentiation Formula (ρ value) |

BBDF : | Block Backward Differentiation Formula of order 3 in [20] |

DI2BBDF : | Diagonally Implicit 2-point BBDF of order 3 in [10] |

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Structure of the A-matrix for fully and diagonally implicit subclasses of the RK methods, given in [16].

k | ${\mathit{\alpha}}_{-2,\mathit{k}}$ | ${\mathit{\alpha}}_{-1,\mathit{k}}$ | ${\mathit{\alpha}}_{0,\mathit{k}}$ | ${\mathit{\alpha}}_{1,\mathit{k}}$ | ${\mathit{\alpha}}_{2,\mathit{k}}$ | ${\mathit{\beta}}_{0,\mathit{k}}$ | ${\mathit{\beta}}_{1,\mathit{k}}$ | ${\mathit{\beta}}_{2,\mathit{k}}$ |
---|---|---|---|---|---|---|---|---|

1 | $\frac{\rho +2}{2\rho -11}$ | $\frac{-3\left(2\rho +3\right)}{2\rho -11}$ | $\frac{3\left(\rho +6\right)}{2\rho -11}$ | 1 | 0 | $\frac{-6\rho}{2\rho -11}$ | $\frac{-6}{2\rho -11}$ | 0 |

2 | $\frac{2\rho +3}{6\rho -19}$ | $\frac{-2\left(3\rho +4\right)}{6\rho -19}$ | 0 | $\frac{-2\left(\rho -12\right)}{6\rho -19}$ | 1 | 0 | $\frac{-12\rho}{6\rho -19}$ | $\frac{-12}{6\rho -19}$ |

$\mathit{\rho}$ or Method | −0.75 | −0.60 | 0.50 | 0.95 | BBDF | DI2BBDF |
---|---|---|---|---|---|---|

$\alpha $ | $85.{657}^{\circ}$ | 86.084${}^{\circ}$ | $88.{352}^{\circ}$ | ${90}^{\circ}$ | $90.{0}^{\circ}$ | $83.{647}^{\circ}$ |

D | −0.156 | −0.115 | −0.016 | 0 | 0 | −0.226 |

$\left|EC\right|$${y}_{n+1}$ | 0.0900 | 0.0984 | 0.1750 | 0.2170 | 0.1667 | 0.1364 |

$\left|EC\right|$${y}_{n+2}$ | 0.1596 | 0.1858 | 0.4688 | 0.6654 | 0.1364 | 0 |

H | Method | MAXE | TIME |
---|---|---|---|

${10}^{-2}$ | $\rho $-DIBBDF(−0.75) | 3.61318 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 7.90031 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

$\rho $-DIBBDF(−0.60) | 3.83043 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 8.90948 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | |

$\rho $-DIBBDF(0.50) | 1.04695 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.90458 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.95) | 1.70999 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 3.15808 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

BBDF | 7.75777 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{8}$ | 9.31533 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | |

DI2BBDF | 3.49466 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.37411 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

${10}^{-4}$ | $\rho $-DIBBDF(−0.75) | 5.14905 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 1.55234 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

$\rho $-DIBBDF(−0.60) | 5.25483 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 2.45657 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.50) | 6.58550 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 2.81742 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.95) | 1.18569 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.92283 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

BBDF | 7.89764 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 8.58038 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

DI2BBDF | 7.77654 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 6.88387 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

${10}^{-6}$ | $\rho $-DIBBDF(−0.75) | 6.28992 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.49431 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

$\rho $-DIBBDF(−0.60) | 6.44415 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.96971 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

$\rho $-DIBBDF(0.50) | 9.41198 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.05194 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

$\rho $-DIBBDF(0.95) | 4.17385 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 3.50504 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

BBDF | 7.89758 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 4.38280 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

DI2BBDF | 6.86129 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.38284 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

H | Method | MAXE | TIME |
---|---|---|---|

${10}^{-2}$ | $\rho $-DIBBDF(−0.75) | 3.02746 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 6.23108 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

$\rho $-DIBBDF(−0.60) | 3.08609 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 9.37159 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | |

$\rho $-DIBBDF(0.50) | 3.79190 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.59550 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.95) | 6.39361 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 3.01500 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

BBDF | 2.27791 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.89023 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

DI2BBDF | 4.51915 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 2.88128 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

${10}^{-4}$ | $\rho $-DIBBDF(−0.75) | 3.97922 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 2.66365 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

$\rho $-DIBBDF(−0.60) | 4.07670 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 3.28285 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.50) | 5.95266 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 6.75064 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.95) | 2.63877 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.72533 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

BBDF | 2.49799 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 4.18826 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

DI2BBDF | 4.33979 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 1.15268 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

${10}^{-6}$ | $\rho $-DIBBDF(−0.75) | 3.99347 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.33363 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

$\rho $-DIBBDF(−0.60) | 4.09109 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.52284 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

$\rho $-DIBBDF(0.50) | 6.00101 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.85521 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

$\rho $-DIBBDF(0.95) | 2.85265 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 2.54802 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

BBDF | 2.49998 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 9.91201 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

DI2BBDF | 4.34093 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.43944 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

H | Method | MAXE | TIME |
---|---|---|---|

${10}^{-2}$ | $\rho $-DIBBDF(−0.75) | 8.78849 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.45541 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

$\rho $-DIBBDF(−0.60) | 9.04698 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.50898 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.50) | 1.13442 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.95526 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.95) | 5.29869 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.24700 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

BBDF | 1.06418 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 8.95481 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

DI2BBDF | 2.25443 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 4.13131 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

${10}^{-4}$ | $\rho $-DIBBDF(−0.75) | 1.58367 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 1.57766 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

$\rho $-DIBBDF(−0.60) | 1.62268 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 1.94590 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

$\rho $-DIBBDF(0.50) | 2.35125 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 2.17491 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

$\rho $-DIBBDF(0.95) | 9.59352 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 3.57073 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

BBDF | 1.11445 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 4.94205 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

DI2BBDF | 1.73985 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 3.35674 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${10}^{-6}$ | $\rho $-DIBBDF(−0.75) | 6.09042 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.53734 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

$\rho $-DIBBDF(−0.60) | 6.20290 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.80857 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

$\rho $-DIBBDF(0.50) | 6.62064 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.11916 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

$\rho $-DIBBDF(0.95) | 4.47822 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 8.19571 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

BBDF | 1.11489 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 1.80581 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

DI2BBDF | 6.30545 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.49118 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

H | Method | MAXE | TIME |
---|---|---|---|

${10}^{-2}$ | $\rho $-DIBBDF(−0.75) | 1.45990 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 3.69631 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

$\rho $-DIBBDF(−0.60) | 1.50371 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 4.54829 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | |

$\rho $-DIBBDF(0.50) | 1.87600 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.29152 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

$\rho $-DIBBDF(0.95) | 2.43046 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.59329 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

BBDF | 1.14580 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{25}$ | 2.03291 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

DI2BBDF | 6.08664 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.34293 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

${10}^{-4}$ | $\rho $-DIBBDF(−0.75) | 5.11045 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.29651 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

$\rho $-DIBBDF(−0.60) | 5.23545 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.70325 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

$\rho $-DIBBDF(0.50) | 7.67139 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.30800 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

$\rho $-DIBBDF(0.95) | 3.40368 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.10669 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |

BBDF | 8.16801 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 2.37062 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

DI2BBDF | 5.55654 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.06007 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

${10}^{-6}$ | $\rho $-DIBBDF(−0.75) | 5.11183 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 1.61469 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

$\rho $-DIBBDF(−0.60) | 5.23685 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 1.63234 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

$\rho $-DIBBDF(0.50) | 7.68199 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 2.65020 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

$\rho $-DIBBDF(0.95) | 3.65574 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 3.65687 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

BBDF | 8.22481 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 6.33621 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | |

DI2BBDF | 5.55636 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 5.27749 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

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**MDPI and ACS Style**

Mohd Ijam, H.; Ibrahim, Z.B.
Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations. *Symmetry* **2019**, *11*, 1342.
https://doi.org/10.3390/sym11111342

**AMA Style**

Mohd Ijam H, Ibrahim ZB.
Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations. *Symmetry*. 2019; 11(11):1342.
https://doi.org/10.3390/sym11111342

**Chicago/Turabian Style**

Mohd Ijam, Hazizah, and Zarina Bibi Ibrahim.
2019. "Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations" *Symmetry* 11, no. 11: 1342.
https://doi.org/10.3390/sym11111342