# Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems

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## Abstract

**:**

## 1. Introduction

## 2. Algebraic Structure of PRK Methods

## 3. Rooted Trees for PRK Methods and Order Conditions

**Example**

**1.**

## 4. Effective Order PRK Methods

## 5. Symplectic PRK Methods with Effective Order 3

#### Derivation of Starting Method

## 6. Mutually Adjoint Symplectic Effective Order PRK Methods

## 7. Numerical Experiments

#### 7.1. Kepler’s Two Body Problem

#### 7.2. Harmonic Oscillator

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Energy error for the Kepler’s Problem ($e=0$) with symplectic effective PRK method using step size $h=2\pi /1000$ for ${10}^{5}$ steps.

**Figure 2.**Energy error for the Kepler’s Problem ($e=0$) with mutually adjoint symplectic effective PRK method using step-size $h=2\pi /1000$ for ${10}^{5}$ steps.

**Figure 3.**Energy error for the Harmonic oscillator problem ($e=0$) with symplectic effective PRK method using step size $h=2\pi /1000$ for ${10}^{5}$ steps.

**Figure 4.**Energy error for the Harmonic oscillator problem ($e=0$) with mutually adjoint symplectic effective PRK method using step size $h=2\pi /1000$ for ${10}^{5}$ steps.

${\mathit{t}}_{\mathit{i}}$ | Tree | $(\mathit{\beta}\mathit{\alpha})({\mathit{t}}_{\mathit{i}})$ | $(\mathit{E}\mathit{\beta})({\mathit{t}}_{\mathit{i}})$ |
---|---|---|---|

${t}_{1}$ | ${\beta}_{1}+{\alpha}_{1}$ | ${\beta}_{1}+1$ | |

${t}_{2}$ | ${\beta}_{2}+{\tilde{\beta}}_{1}{\alpha}_{1}+{\alpha}_{2}$ | ${\beta}_{2}+{\beta}_{1}+{\displaystyle \frac{1}{2}}$ | |

${t}_{3}$ | ${\beta}_{3}+{\tilde{\beta}}_{1}^{2}{\alpha}_{1}+2{\tilde{\beta}}_{1}{\alpha}_{2}+{\alpha}_{3}$ | ${\beta}_{3}+2{\beta}_{2}+{\beta}_{1}+{\displaystyle \frac{1}{3}}$ | |

${t}_{4}$ | ${\beta}_{4}+{\beta}_{1}{\alpha}_{2}+{\tilde{\beta}}_{2}{\alpha}_{1}+{\alpha}_{4}$ | ${\beta}_{4}+{\beta}_{2}+{\displaystyle \frac{1}{2}}{\beta}_{1}+{\displaystyle \frac{1}{6}}$ |

${\mathit{t}}_{\mathit{i}}$ | Tree | $(\tilde{\mathit{\beta}}\tilde{\mathit{\alpha}})({\mathit{t}}_{\mathit{i}})$ | $(\mathit{E}\tilde{\mathit{\beta}})({\mathit{t}}_{\mathit{i}})$ |
---|---|---|---|

${\tilde{t}}_{1}$ | ${\tilde{\beta}}_{1}+{\tilde{\alpha}}_{1}$ | ${\tilde{\beta}}_{1}+1$ | |

${\tilde{t}}_{2}$ | ${\tilde{\beta}}_{2}+{\beta}_{1}{\tilde{\alpha}}_{1}+{\tilde{\alpha}}_{2}$ | ${\tilde{\beta}}_{2}+{\tilde{\beta}}_{1}+{\displaystyle \frac{1}{2}}$ | |

${\tilde{t}}_{3}$ | ${\tilde{\beta}}_{3}+{\beta}_{1}^{2}{\tilde{\alpha}}_{1}+2{\beta}_{1}{\tilde{\alpha}}_{2}+{\tilde{\alpha}}_{3}$ | ${\tilde{\beta}}_{3}+2{\tilde{\beta}}_{2}+{\tilde{\beta}}_{1}+{\displaystyle \frac{1}{3}}$ | |

${\tilde{t}}_{4}$ | ${\tilde{\beta}}_{4}+{\tilde{\beta}}_{1}{\tilde{\alpha}}_{2}+{\beta}_{2}{\tilde{\alpha}}_{1}+{\tilde{\alpha}}_{4}$ | ${\tilde{\beta}}_{4}+{\tilde{\beta}}_{2}+{\displaystyle \frac{1}{2}}{\tilde{\beta}}_{1}+{\displaystyle \frac{1}{6}}$ |

t | Elementary Differentials | $\mathbf{\Phi}\left(\mathit{t}\right)$ | $\tilde{\mathit{t}}$ | Elementary Differentials | $\mathbf{\Phi}\left(\tilde{\mathit{t}}\right)$ |
---|---|---|---|---|---|

k | ${b}_{i}$ | r | ${\tilde{b}}_{i}$ | ||

$\frac{\partial k}{\partial v}}r$ | ${b}_{i}{\tilde{c}}_{i}$ | $\frac{\partial r}{\partial u}}k$ | ${\tilde{b}}_{i}{c}_{i}$ | ||

$\frac{{\partial}^{2}k}{\partial v\phantom{\rule{3.33333pt}{0ex}}\partial v}}(r,r)$ | ${b}_{i}{\tilde{c}}_{i}^{2}$ | $\frac{{\partial}^{2}r}{\partial u\phantom{\rule{3.33333pt}{0ex}}\partial u}}(k,k)$ | ${\tilde{b}}_{i}{c}_{i}^{2}$ | ||

$\frac{\partial k}{\partial v}}{\displaystyle \frac{\partial r}{\partial u}}k$ | ${b}_{i}{\tilde{a}}_{ij}{c}_{j}$ | $\frac{\partial r}{\partial u}}{\displaystyle \frac{\partial k}{\partial v}}r$ | ${\tilde{b}}_{i}{a}_{ij}{\tilde{c}}_{j}$ |

Tree | No Cut | First Cut | Second Cut | Third Cut | All Cuts |
---|---|---|---|---|---|

t | |||||

u | |||||

$t\backslash u$ | |||||

term | ${\beta}_{3}$ | ${\tilde{\beta}}_{1}{\alpha}_{2}$ | ${\tilde{\beta}}_{1}{\alpha}_{3}$ | ${\tilde{\beta}}_{1}^{2}{\alpha}_{1}$ | ${\alpha}_{3}$ |

h | n | Global Error | Ratio |
---|---|---|---|

$\frac{\pi}{225}$ | 225 | $7.7741637102284\times {10}^{-04}$ | |

$8.927465$ | |||

$\frac{\pi}{450}$ | 450 | $8.7081425338124\times {10}^{-05}$ | |

$8.475956$ | |||

$\frac{\pi}{900}$ | 900 | $1.02739357736254\times {10}^{-05}$ | |

$8.063877$ | |||

$\frac{\pi}{1800}$ | 1800 | $1.27406905909534\times {10}^{-06}$ |

**Table 6.**Global errors and their comparison: Mutually adjoint symplectic Effective order PRK method.

h | n | Global Error | Ratio |
---|---|---|---|

$\frac{\pi}{40}$ | 40 | $4.635890382086\times {10}^{-03}$ | |

$7.963129$ | |||

$\frac{\pi}{80}$ | 80 | $5.82169473987045\times {10}^{-04}$ | |

$8.062262$ | |||

$\frac{\pi}{160}$ | 160 | $7.22091971134495\times {10}^{-05}$ | |

$7.836579$ | |||

$\frac{\pi}{320}$ | 320 | $9.21437776243649\times {10}^{-06}$ |

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**MDPI and ACS Style**

Ahmad, J.; Habib, Y.; Rehman, S.u.; Arif, A.; Shafiq, S.; Younas, M.
Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems. *Symmetry* **2019**, *11*, 142.
https://doi.org/10.3390/sym11020142

**AMA Style**

Ahmad J, Habib Y, Rehman Su, Arif A, Shafiq S, Younas M.
Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems. *Symmetry*. 2019; 11(2):142.
https://doi.org/10.3390/sym11020142

**Chicago/Turabian Style**

Ahmad, Junaid, Yousaf Habib, Shafiq ur Rehman, Azqa Arif, Saba Shafiq, and Muhammad Younas.
2019. "Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems" *Symmetry* 11, no. 2: 142.
https://doi.org/10.3390/sym11020142