# On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension

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## Abstract

**:**

## 1. Introduction

## 2. One-Step Symmetric Hermite–Obreshkov Methods

**Lemma**

**1.**

**Corollary**

**1.**

## 3. Conjugate Symplecticity of the Symmetric One-Step BSHO Methods

**Lemma**

**2.**

- (a)
- ${\mathsf{\Phi}}_{h}(\mathbf{y})$ has a modified first integral of the form $\tilde{Q}(\mathbf{y})=Q(\mathbf{y})+h{Q}_{1}(\mathbf{y})+{h}^{2}{Q}_{2}(\mathbf{y})+...$ where each ${Q}_{i}(\xb7)$ is a differential functional;
- (b)
- ${\mathsf{\Phi}}_{h}(\mathbf{y})$ is conjugate to a symplectic B-series integrator.

**Theorem**

**1.**

**Proof.**

## 4. The Spline Extension

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 4.1. Efficient Spline Computation

#### 4.2. Spline Convergence

**Proposition**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Approximation of the Derivatives

- using generic symbolic tools, if the function $\mathbf{f}$ is known in closed form;
- using a tool of automatic differentiation, like ADiGator, a MATLAB Automatic Differentiation Tool [24];
- approximating it with, for example, finite differences.

**Remark**

**1.**

## 6. Numerical Examples

#### 6.1. Kepler Problem

#### 6.2. Non-Linear Pendulum Problem

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sparsity structure of the matrix A with $N=8$, $R=1$ (

**left**) and with $N=8$, $R=2$ (

**right**).

**Figure 2.**Kepler problem: results for the sixth (BSHO6, red dotted line) and eighth (BSHO8, purple dotted line) order BSHO methods, sixth order Euler–Maclaurin method (EMHO6, blue solid line) and sixth (Gauss–Runge–Kutta (GRK6), yellow dashed line) and eighth (GRK8-green dashed line) order Gauss methods. (

**Top-left**) Absolute error of the numerical solution; (

**top-right**) error in the Hamiltonian function; (

**bottom-left**) error in the angular momentum; (

**bottom-right**) error in the second component of the Lenz vector.

**Figure 3.**Nonlinear pendulum problem: results for the Hermite–Obreshkov method of order six and eight (BSHO6, red, and BSHO8, purple dotted lines), for the sixth order Euler–Maclaurin (EMHO6, blue solid line) and Gauss methods (GRK6, yellow, and GRK8, green dashed lines) applied to the pendulum problem. (

**Left**) plot: absolute error of the numerical solution; (

**upper-right**) and (

**bottom-right**) plots: error in the Hamiltonian function for the sixth order and eighth order integrators, respectively.

R | ${\mathit{\beta}}_{1}^{(\mathit{R})}$ | ${\mathit{\beta}}_{2}^{(\mathit{R})}$ | ${\mathit{\beta}}_{3}^{(\mathit{R})}$ | ${\mathit{\beta}}_{4}^{(\mathit{R})}$ | ${\mathit{\beta}}_{5}^{(\mathit{R})}$ |
---|---|---|---|---|---|

1 | $\frac{1}{2}$ | ||||

2 | $\frac{1}{2}$ | $\frac{1}{12}$ | |||

3 | $\frac{1}{2}$ | $\frac{1}{10}$ | $\frac{1}{120}$ | ||

4 | $\frac{1}{2}$ | $\frac{3}{28}$ | $\frac{1}{84}$ | $\frac{1}{1680}$ | |

5 | $\frac{1}{2}$ | $\frac{1}{9}$ | $\frac{1}{72}$ | $\frac{1}{1008}$ | $\frac{1}{30240}$ |

R | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${\delta}_{R}$ | $\frac{{b}_{2}}{2!}$ | $\frac{{b}_{4}}{4!}$ | $\frac{3}{10}\frac{{b}_{6}}{6!}$ | $\frac{1}{21}\frac{{b}_{8}}{8!}$ | $\frac{1}{210}\frac{{b}_{10}}{10!}$ |

${b}_{2R}$ | $\frac{1}{6}$ | $-\frac{1}{30}$ | $\frac{1}{42}$ | $-\frac{1}{30}$ | $\frac{5}{66}$ |

**Table 3.**Kepler problem: maximum absolute error of the numerical solution and its derivative computed for 10 periods.

Order | N | erry | Rate | erry | Rate |
---|---|---|---|---|---|

4 | 100 | $2.69\xb7{10}^{-1}$ | $1.33\xb7{10}^{0}$ | ||

4 | 200 | $1.69\xb7{10}^{-2}$ | 3.99 | $8.50\xb7{10}^{-2}$ | 3.96 |

4 | 400 | $1.06\xb7{10}^{-3}$ | 4.00 | $5.30\xb7{10}^{-3}$ | 4.00 |

4 | 800 | $6.60\xb7{10}^{-5}$ | 4.00 | $3.31\xb7{10}^{-4}$ | 4.00 |

6 | 100 | $1.95\xb7{10}^{-3}$ | $9.74\xb7{10}^{-3}$ | ||

6 | 200 | $2.96\xb7{10}^{-5}$ | 6.03 | $1.48\xb7{10}^{-4}$ | 6.03 |

6 | 400 | $4.60\xb7{10}^{-7}$ | 6.00 | $2.30\xb7{10}^{-6}$ | 6.00 |

6 | 800 | $7.19\xb7{10}^{-9}$ | 6.00 | $3.60\xb7{10}^{-8}$ | 6.00 |

8 | 100 | $1.56\xb7{10}^{-5}$ | $7.82\xb7{10}^{-5}$ | ||

8 | 200 | $5.75\xb7{10}^{-8}$ | 8.08 | $2.88\xb7{10}^{-7}$ | 8.08 |

8 | 400 | $2.17\xb7{10}^{-10}$ | 8.05 | $1.08\xb7{10}^{-9}$ | 8.05 |

8 | 800 | $7.62\xb7{10}^{-12}$ | 4.87 | $3.70\xb7{10}^{-11}$ | 4.44 |

**Table 4.**Nonlinear pendulum problem: Maximum absolute error of the numerical solution and its derivative computed for 10 periods.

Order | N | erry | Rate | erry | Rate |
---|---|---|---|---|---|

4 | 10 | $1.26\xb7{10}^{-2}$ | $1.28\xb7{10}^{-2}$ | ||

4 | 20 | $9.02\xb7{10}^{-4}$ | 3.81 | $1.10\xb7{10}^{-3}$ | 3.53 |

4 | 40 | $5.73\xb7{10}^{-5}$ | 3.97 | $6.60\xb7{10}^{-5}$ | 4.06 |

4 | 80 | $3.58\xb7{10}^{-6}$ | 4.00 | $4.52\xb7{10}^{-6}$ | 3.86 |

6 | 10 | $2.65\xb7{10}^{-4}$ | $2.82\xb7{10}^{-4}$ | ||

6 | 20 | $1.36\xb7{10}^{-6}$ | 7.59 | $5.77\xb7{10}^{-6}$ | 5.61 |

6 | 40 | $2.07\xb7{10}^{-8}$ | 6.04 | $1.15\xb7{10}^{-8}$ | 5.65 |

6 | 80 | $3.21\xb7{10}^{-10}$ | 6.01 | $1.81\xb7{10}^{-9}$ | 5.98 |

8 | 10 | $2.56\xb7{10}^{-5}$ | $2.61\xb7{10}^{-5}$ | ||

8 | 20 | $1.53\xb7{10}^{-8}$ | 10.7 | $8.50\xb7{10}^{-8}$ | 8.26 |

8 | 40 | $6.14\xb7{10}^{-11}$ | 7.96 | $4.02\xb7{10}^{-10}$ | 7.72 |

8 | 80 | $3.01\xb7{10}^{-13}$ | 7.67 | $1.56\xb7{10}^{-12}$ | 8.01 |

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**MDPI and ACS Style**

Mazzia, F.; Sestini, A. On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension. *Axioms* **2018**, *7*, 58.
https://doi.org/10.3390/axioms7030058

**AMA Style**

Mazzia F, Sestini A. On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension. *Axioms*. 2018; 7(3):58.
https://doi.org/10.3390/axioms7030058

**Chicago/Turabian Style**

Mazzia, Francesca, and Alessandra Sestini. 2018. "On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension" *Axioms* 7, no. 3: 58.
https://doi.org/10.3390/axioms7030058