# Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

- i
- $f$is non-negative and non-decreasing in each$x,{s}_{0},{s}_{1},\dots ,{s}_{n-1}$in$\Re $,
- ii
- $f\left(x,{c}_{0},{c}_{1},\dots ,{c}_{n-1}\right)>0$for${x}_{0}\le x\le {x}_{0}+a$, and
- iii
- ${c}_{k}\ge 0,\text{}k=0,1,\dots ,n-1$.

**Proof of Theorem 1**, see [2].

## 2. Derivation of the NFSSA

## 3. Validation of Properties of the NFSSA

#### 3.1. Order and Error Constant

**Definition**

**1.**

#### 3.2. Zero-Stability

**Definition**

**2.**

#### 3.3. Consistency

#### 3.4. Convergence

**Theorem**

**2.**

**Proof of Theorem 2**, see [41].

#### 3.5. Regions of Absolute Stability

**Definition**

**4.**

## 4. Pseudocode and Step-Size Selection for Implementation of the NFSSA

#### 4.1. Pseudocode

#### 4.2. Step-Size Selection

## 5. Numerical Examples

#### 5.1. Problem 1

#### 5.2. Problem 2

#### 5.3. Problem 3

#### 5.4. Problem 4

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

h | Step size |

NST | Number of steps taken |

FNE | Number of function evaluations |

FNC | Total number of function calls |

FLS | Number of failure (rejected) steps |

EXECTM | Execution time (in seconds) |

ABERR | Absolute error |

MAXERR | Maximum error |

APPSOL | Approximate solution |

VSSM | Order 6 variable step-size method developed by [1] |

HSDBBDF | Order 7 hybrid second derivative block backward differentiation formula developed by [11] |

SDM | Order 11 second derivative method developed by [50] |

ode 15s | MATLAB inbuilt stiff solver |

NFSSA | Newly derived non-fixed step-size algorithm |

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Step-Size | Formulae |
---|---|

Ratio | |

$r=1$ | ${y}_{n+\frac{1}{2}}={y}_{n}+h\left(\frac{23}{\mathrm{112,896}}{f}_{n-2}-\frac{419}{\mathrm{120,960}}{f}_{n-1}+\frac{2137}{\mathrm{10,080}}{f}_{n}+\frac{2689}{7560}{f}_{n+\frac{1}{2}}-\frac{3407}{\mathrm{40,320}}{f}_{n+1}+\frac{407}{\mathrm{17,640}}{f}_{n+\frac{3}{2}}-\frac{727}{\mathrm{241,920}}{f}_{n+2}\right)$ |

${y}_{n+1}={y}_{n}+h\left(\frac{1}{\mathrm{11,760}}{f}_{n-2}-\frac{13}{7560}{f}_{n-1}+\frac{19}{105}{f}_{n}+\frac{604}{945}{f}_{n+\frac{1}{2}}+\frac{157}{840}{f}_{n+1}-\frac{4}{735}{f}_{n+\frac{3}{2}}+\frac{1}{\mathrm{15,120}}{f}_{n+2}\right)$ | |

${y}_{n+\frac{3}{2}}={y}_{n}+h\left(\frac{3}{\mathrm{12,544}}{f}_{n-2}-\frac{17}{4480}{f}_{n-1}+\frac{117}{560}{f}_{n}+\frac{151}{280}{f}_{n+\frac{1}{2}}+\frac{2481}{4480}{f}_{n+1}+\frac{411}{1960}{f}_{n+\frac{3}{2}}-\frac{73}{8960}{f}_{n+2}\right)$ | |

${y}_{n+2}={y}_{n}+h\left(-\frac{1}{4410}{f}_{n-2}+\frac{2}{945}{f}_{n-1}+\frac{44}{315}{f}_{n}+\frac{704}{945}{f}_{n+\frac{1}{2}}+\frac{74}{315}{f}_{n+1}+\frac{320}{441}{f}_{n+\frac{3}{2}}+\frac{289}{1890}{f}_{n+2}\right)$ | |

$r=2$ | ${y}_{n+\frac{1}{2}}={y}_{n}+h\left(\frac{419}{\mathrm{31,933,440}}{f}_{n-2}-\frac{797}{\mathrm{2,257,920}}{f}_{n-1}+\frac{\mathrm{62,099}}{\mathrm{322,560}}{f}_{n}+\frac{1781}{4536}{f}_{n+\frac{1}{2}}-\frac{583}{5040}{f}_{n+1}+\frac{6991}{\mathrm{194,040}}{f}_{n+\frac{3}{2}}-\frac{5003}{\mathrm{967,680}}{f}_{n+2}\right)$ |

${y}_{n+1}={y}_{n}+h\left(\frac{13}{\mathrm{1,995,840}}{f}_{n-2}-\frac{3}{\mathrm{15,680}}{f}_{n-1}+\frac{1151}{6720}{f}_{n}+\frac{1864}{2835}{f}_{n+\frac{1}{2}}+\frac{6}{35}{f}_{n+1}+\frac{8}{8085}{f}_{n+\frac{3}{2}}-\frac{61}{\mathrm{60,480}}{f}_{n+2}\right)$ | |

${y}_{n+\frac{3}{2}}={y}_{n}+h\left(\frac{17}{\mathrm{1,182,720}}{f}_{n-2}-\frac{93}{\mathrm{250,880}}{f}_{n-1}+\frac{6723}{\mathrm{35,840}}{f}_{n}+\frac{487}{840}{f}_{n+\frac{1}{2}}+\frac{291}{560}{f}_{n+1}+\frac{4827}{\mathrm{21,560}}{f}_{n+\frac{3}{2}}-\frac{377}{\mathrm{35,840}}{f}_{n+2}\right)$ | |

${y}_{n+2}={y}_{n}+h\left(-\frac{1}{\mathrm{124,740}}{f}_{n-2}+\frac{1}{8820}{f}_{n-1}+\frac{191}{1260}{f}_{n}+\frac{2048}{2835}{f}_{n+\frac{1}{2}}+\frac{16}{63}{f}_{n+1}+\frac{\mathrm{17,408}}{\mathrm{24,255}}{f}_{n+\frac{3}{2}}+\frac{583}{3780}{f}_{n+2}\right)$ | |

$r=\frac{1}{2}$ | ${y}_{n+\frac{1}{2}}={y}_{n}+h\left(\frac{271}{\mathrm{120,960}}{f}_{n-2}-\frac{23}{1008}{f}_{n-1}+\frac{\mathrm{10,273}}{\mathrm{40,320}}{f}_{n}+\frac{293}{945}{f}_{n+\frac{1}{2}}-\frac{2257}{\mathrm{40,320}}{f}_{n+1}+\frac{67}{5040}{f}_{n+\frac{3}{2}}-\frac{191}{\mathrm{120,960}}{f}_{n+2}\right)$ |

${y}_{n+1}={y}_{n}+h\left(\frac{1}{1512}{f}_{n-2}-\frac{1}{105}{f}_{n-1}+\frac{167}{840}{f}_{n}+\frac{586}{945}{f}_{n+\frac{1}{2}}+\frac{167}{840}{f}_{n+1}-\frac{1}{105}{f}_{n+\frac{3}{2}}+\frac{1}{1512}{f}_{n+2}\right)$ | |

${y}_{n+\frac{3}{2}}={y}_{n}+h\left(\frac{13}{4480}{f}_{n-2}-\frac{3}{112}{f}_{n-1}+\frac{1161}{4480}{f}_{n}+\frac{17}{35}{f}_{n+\frac{1}{2}}+\frac{2631}{4480}{f}_{n+1}+\frac{111}{560}{f}_{n+\frac{3}{2}}-\frac{29}{4480}{f}_{n+2}\right)$ | |

${y}_{n+2}={y}_{n}+h\left(-\frac{4}{945}{f}_{n-2}+\frac{8}{315}{f}_{n-1}+\frac{29}{315}{f}_{n}+\frac{752}{945}{f}_{n+\frac{1}{2}}+\frac{64}{315}{f}_{n+1}+\frac{323}{315}{f}_{n+\frac{3}{2}}+\frac{143}{945}{f}_{n+2}\right)$ |

Step-Size Ratio | Predictor Formulae |
---|---|

$r=1$ | ${y}_{n+\frac{1}{2}}^{p}={y}_{n}+\frac{h}{12}\left({f}_{n-2}-\frac{7}{2}{f}_{n-1}+\frac{17}{2}{f}_{n}\right)$ |

${y}_{n+1}^{p}={y}_{n}+\frac{h}{3}\left(\frac{5}{4}{f}_{n-2}-4{f}_{n-1}+\frac{23}{4}{f}_{n}\right)$ | |

${y}_{n+\frac{3}{2}}^{p}={y}_{n}+\frac{h}{4}\left(\frac{9}{2}{f}_{n-2}-\frac{27}{2}{f}_{n-1}+15{f}_{n}\right)$ | |

${y}_{n+2}^{p}={y}_{n}+\frac{h}{3}\left(7{f}_{n-2}-20{f}_{n-1}+19{f}_{n}\right)$ | |

$r=2$ | ${y}_{n+\frac{1}{2}}^{p}={y}_{n}+\frac{h}{96}\left(\frac{7}{2}{f}_{n-2}-13{f}_{n-1}+\frac{115}{2}{f}_{n}\right)$ |

${y}_{n+1}^{p}={y}_{n}+\frac{h}{6}\left({f}_{n-2}-\frac{7}{2}{f}_{n-1}+\frac{17}{2}{f}_{n}\right)$ | |

${y}_{n+\frac{3}{2}}^{p}={y}_{n}+\frac{h}{32}\left(\frac{27}{2}{f}_{n-2}-45{f}_{n-1}+\frac{159}{2}{f}_{n}\right)$ | |

${y}_{n+2}^{p}={y}_{n}+\frac{h}{3}\left(\frac{5}{2}{f}_{n-2}-8{f}_{n-1}+\frac{23}{2}{f}_{n}\right)$ | |

$r=\frac{1}{2}$ | ${y}_{n+\frac{1}{2}}^{p}={y}_{n}+\frac{h}{3}\left(\frac{5}{8}{f}_{n-2}-2{f}_{n-1}+\frac{23}{8}{f}_{n}\right)$ |

${y}_{n+1}^{p}={y}_{n}+\frac{h}{3}\left(\frac{7}{2}{f}_{n-2}-10{f}_{n-1}+\frac{19}{2}{f}_{n}\right)$ | |

${y}_{n+\frac{3}{2}}^{p}={y}_{n}+\frac{h}{8}\left(27{f}_{n-2}-72{f}_{n-1}+57{f}_{n}\right)$ | |

${y}_{n+2}^{p}={y}_{n}+\frac{h}{3}\left(22{f}_{n-2}-56{f}_{n-1}+40{f}_{n}\right)$ |

Step-Size Ratio | Roots |
---|---|

$r=1$ | ${t}_{1}={t}_{2}={t}_{3}=0,{t}_{4}=1$ |

$r=2$ | ${t}_{1}={t}_{2}={t}_{3}=0,{t}_{4}=1$ |

$r=1/2$ | ${t}_{1}={t}_{2}={t}_{3}=0,{t}_{4}=1$ |

Step-Size Ratio | Stability Interval |
---|---|

$r=1$ | $\left(-9.6,0\right)$ |

$r=2$ | $\left(-17.2,0\right)$ |

$r=1/2$ | $\left(-5.0,0\right)$ |

h_{initial} | TOL | Method | MAXERR (y_{1}) | MAXERR (y_{2}) | MAXERR (y_{3}) | NST | FNE | FLS | EXECT |
---|---|---|---|---|---|---|---|---|---|

10^{−}^{7} | 10^{−}^{10} | NFSSA | 4.1983 × 10^{−}^{19} | 3.1041 × 10^{−}^{23} | 5.0013 × 10^{−}^{19} | 3902 | 7110 | 00 | 9.05 |

VSSM | 4.4408 × 10^{−}^{16} | 3.3881 × 10^{−}^{21} | 5.5511 × 10^{−}^{17} | 3730 | 14,920 | 00 | 13.88 | ||

ode 15s | 7.7561 × 10^{−}^{9} | 5.4664 × 10^{−}^{12} | 8.2009 × 10^{−}^{10} | 2006 | 56,090 | 00 | 43.98 | ||

10^{−}^{8} | 10^{−}^{11} | NFSSA | 6.3923 × 10^{−}^{19} | 1.1680 × 10^{−}^{23} | 3.0361 × 10^{−}^{19} | 6960 | 16,780 | 00 | 19.12 |

VSSM | 8.8817 × 10^{−}^{16} | 1.8634 × 10^{−}^{20} | 4.9960 × 10^{−}^{16} | 6626 | 26,504 | 00 | 24.46 | ||

ode 15s | 9.5637 × 10^{−}^{9} | 7.9259 × 10^{−}^{12} | 8.9771 × 10^{−}^{10} | 4344 | 60,020 | 00 | 68.46 | ||

10^{−}^{9} | 10^{−}^{12} | NFSSA | 2.5256 × 10^{−}^{19} | 3.7445 × 10^{−}^{22} | 1.0927 × 10^{−}^{18} | 12,002 | 28,080 | 00 | 34.81 |

VSSM | 3.2196 × 10^{−}^{15} | 3.2187 × 10^{−}^{20} | 2.7755 × 10^{−}^{15} | 11,766 | 47,064 | 00 | 45.10 | ||

ode 15s | 6.0929 × 10^{−}^{9} | 4.3587 × 10^{−}^{12} | 4.7460 × 10^{−}^{10} | 7608 | 64,030 | 00 | 82.22 | ||

10^{−}^{10} | NFSSA | 6.1523 × 10^{−}^{18} | 4.1914 × 10^{−}^{22} | 9.2440 × 10^{−}^{17} | 21,228 | 56,024 | 00 | 69.02 | |

VSSM | 4.3298 × 10^{−}^{15} | 2.3716 × 10^{−}^{20} | 8.3266 × 10^{−}^{16} | 20,916 | 83,664 | 00 | 84.63 | ||

ode 15s | 7.2701 × 10^{−}^{9} | 6.2401 × 10^{−}^{12} | 6.4396 × 10^{−}^{10} | 15,018 | 72,670 | 00 | 103.47 |

X | y_{i} | SDM | ode 15s | HSDBBDF | NFSSA |
---|---|---|---|---|---|

5 | y_{1} | 1.9196 × 10^{−}^{02} | 5.4044 × 10^{−}^{12} | 1.0123 × 10^{−}^{17} | 3.0145 × 10^{−}^{19} |

y_{2} | 3.1501 × 10^{−}^{2} | 7.4701 × 10^{−}^{8} | 3.3881 × 10^{−}^{21} | 5.5511 × 10^{−}^{17} | |

40 | y_{1} | 7.9295 × 10^{−}^{10} | 1.9841 × 10^{−}^{13} | 5.8776 × 10^{−}^{18} | 7.7121 × 10^{−}^{21} |

y_{2} | 9.7003 × 10^{−}^{6} | 3.9544 × 10^{−}^{9} | 7.8381 × 10^{−}^{10} | 5.2001 × 10^{−}^{13} | |

70 | y_{1} | 1.8759 × 10^{−}^{13} | 2.2934 × 10^{−}^{14} | 4.0276 × 10^{−}^{18} | 4.1291 × 10^{−}^{21} |

y_{2} | 9.1528 × 10^{−}^{8} | 7.0561 × 10^{−}^{10} | 5.3701 × 10^{−}^{10} | 3.9866 × 10^{−}^{13} | |

100 | y_{1} | 1.7835 × 10^{−}^{18} | 6.4358 × 10^{−}^{16} | 5.3649 × 10^{−}^{19} | 1.9087 × 10^{−}^{21} |

y_{2} | 4.4647 × 10^{−}^{10} | 3.0223 × 10^{−}^{10} | 7.1532 × 10^{−}^{11} | 1.2781 × 10^{−}^{13} |

NST | Method | ABERR (y _{1}) | ABERR (y _{2}) | FNE | FLS | EXECTM |
---|---|---|---|---|---|---|

44 | NFSSA | 6.4244 × 10^{−}^{15} | 8.1479 × 10^{−}^{14} | 156 | 00 | 0.0781 |

VSSM | 3.3229 × 10^{−}^{11} | 7.5785 × 10^{−}^{11} | 176 | 00 | 0.1248 | |

ode 15s | 2.2775 × 10^{−}^{9} | 5.9313 × 10^{−}^{9} | 198 | 00 | 0.1934 | |

46 | NFSSA | 7.1241 × 10^{−}^{14} | 9.7146 × 10^{−}^{14} | 164 | 00 | 0.0899 |

VSSM | 5.7434 × 10^{−}^{11} | 8.4486 × 10^{−}^{10} | 184 | 00 | 0.1404 | |

ode 15s | 4.0301 × 10^{−}^{9} | 8.3121 × 10^{−}^{9} | 208 | 00 | 0.2018 | |

50 | NFSSA | 4.6218 × 10^{−}^{12} | 5.0121 × 10^{−}^{12} | 180 | 00 | 0.0914 |

VSSM | 2.5632 × 10^{−}^{10} | 3.9818 × 10^{−}^{10} | 200 | 00 | 0.1440 | |

ode 15s | 7.1131 × 10^{−}^{9} | 1.5736 × 10^{−}^{8} | 228 | 00 | 0.2186 |

X | y_{i} | APPSOL Using NFSSA | APPSOL Using ode 15s |
---|---|---|---|

1 | y_{1} | −1.8650950986 | −1.8650950571 |

y_{2} | 0.7524845366 | 0.7524845299 | |

5 | y_{1} | 1.8985234725 | 1.8985234421 |

y_{2} | −0.7289532611 | −0.7289532451 | |

10 | y_{1} | 1.7865365303 | 1.7865365103 |

y_{2} | −0.8156276699 | −0.8156276438 | |

20 | y_{1} | 1.5075643350 | 1.5075643177 |

y_{2} | −1.1911230102 | −1.1911230003 |

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Sunday, J.; Shokri, A.; Kwanamu, J.A.; Nonlaopon, K.
Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy. *Symmetry* **2022**, *14*, 1575.
https://doi.org/10.3390/sym14081575

**AMA Style**

Sunday J, Shokri A, Kwanamu JA, Nonlaopon K.
Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy. *Symmetry*. 2022; 14(8):1575.
https://doi.org/10.3390/sym14081575

**Chicago/Turabian Style**

Sunday, Joshua, Ali Shokri, Joshua Amawa Kwanamu, and Kamsing Nonlaopon.
2022. "Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy" *Symmetry* 14, no. 8: 1575.
https://doi.org/10.3390/sym14081575