# Temperature Error Reduction of DPD Fluid by Using Partitioned Runge-Kutta Time Integration Scheme

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equations

#### 2.2. Time Integration Algorithms

#### Partitioned Runge-Kutta Scheme

## 3. Results and Discussion

#### 3.1. Simulation Details

#### 3.2. Temperature Errors on Different Time Integration Schemes

#### 3.3. Errors in Configurational and Dynamic Properties

#### 3.4. Computational Efficiency

## 4. Conclusions

- The comparison of the temperature errors among three PRK3 schemes showed that the PRK3 (Ruth) scheme can be superior to the existing schemes in both kinetic and configurational temperatures. Whereas the PRK3 (Iwatsu A) scheme has a disadvantage in the kinetic temperature error. Both temperature errors obtained by using the PRK3 (Iwatsu B) scheme were inferior to those by the existing scheme. This series of results are almost ranked in the same order of computational error and degree of dispersibility shown in the previous study though the time integrations of the non-conservative part for these PRK3 schemes differently influence on the error as well as the conservative part. This strongly supports the findings in our previous study that the conservative part of the PSP is significantly influences the time integration in the DPD simulations.
- The radial distribution function and velocity autocorrelation function were estimated for the simulations by using the existing and PRK3 schemes to compare the errors in configurational and kinetic quantities. It was found from this comparison that the PRK3 (Ruth) scheme can be regarded as one of the best scheme among the available schemes.
- Comparison of kinetic/configurational quantities and temperatures showed that the kinetic and configurational temperatures involve errors from different characteristics. Thus, both temperatures are important to scale the computational error in DPD.
- The computational efficiency was finally estimated. The results showed that the PRK3 (Ruth) scheme is more efficient than the existing schemes, and substantially low error can be kept in a wide time increment range.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Numerical Time Integration Schemes

#### Appendix A.1. M-Verlet Scheme

#### Appendix A.2. Shardlow and M-Shardlow Schemes

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**Figure 1.**Errors in (

**a**) ${\left({k}_{B}T\right)}_{K}$ and (

**b**) ${\left({k}_{B}T\right)}_{C}$ as a function of time increment for different time integration schemes.

**Figure 2.**Errors in (

**a**) ${\left({k}_{B}T\right)}_{K}$ and (

**b**) ${\left({k}_{B}T\right)}_{C}$ as a function of time increment for different PRK3 schemes.

**Figure 3.**Radial distribution function on different time integration schemes for (

**a**) $\Delta t$ = 0.01 and (

**b**) $\Delta t$ = 0.1. The solid line is the result of the M-Verlet scheme for $\Delta t=0.005$.

**Figure 4.**Error in potential energy calculated using Equation (22) as a function of $\Delta t$ on different time integration schemes.

**Figure 5.**Velocity autocorrelation function as a function of delay time on different time integration schemes for (

**a**) $\Delta t$ = 0.005 and (

**b**) $\Delta t$ = 0.1.

**Figure 6.**Error in diffusion coefficient calculated using Equation (26) as a function of $\Delta t$ on different time integration schemes.

Ruth [24] | Iwatsu A [25] | Iwatsu B [25] | |
---|---|---|---|

${b}_{1}$ | $\frac{7}{24}$ | $\frac{-7+\sqrt{209/2}}{12}$ | $-\frac{7+\sqrt{209/2}}{12}$ |

${b}_{2}$ | $\frac{3}{4}$ | $\frac{11}{12}$ | $\frac{11}{12}$ |

${b}_{3}$ | $-\frac{1}{24}$ | $\frac{8-\sqrt{209/2}}{12}$ | $\frac{8+\sqrt{209/2}}{12}$ |

${\widehat{b}}_{1}$ | $\frac{2}{3}$ | $\frac{2}{9}\left(1+\sqrt{\frac{38}{11}}\right)$ | $\frac{2}{9}\left(1-\sqrt{\frac{38}{11}}\right)$ |

${\widehat{b}}_{2}$ | $-\frac{2}{3}$ | $\frac{2}{9}\left(1-\sqrt{\frac{38}{11}}\right)$ | $\frac{2}{9}\left(1+\sqrt{\frac{38}{11}}\right)$ |

${\widehat{b}}_{3}$ | 1 | $\frac{5}{9}$ | $\frac{5}{9}$ |

**Table 2.**Scaled-efficiency for the M-Verlet, M-Shardlow, and PRK3 (Ruth) schemes. The efficiencies are normalized by the one for the M-Verlet scheme. (TH: Threshold).

Scheme | 10% TH | 1% TH | ||
---|---|---|---|---|

$\mathbf{\Delta}{\mathit{t}}_{\mathit{C}}$ | SE${}_{0}$ | $\mathbf{\Delta}{\mathit{t}}_{\mathit{C}}$ | SE${}_{0}$ | |

M-Verlet | 0.05 | 1 | 0.01 | 1 |

M-Shardlow | 0.1 | 0.39 | 0.02 | 0.40 |

PRK3 (Ruth) | 0.1 | 1.02 | 0.05 | 2.55 |

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**MDPI and ACS Style**

Yamada, T.; Itoh, S.; Morinishi, Y.; Tamano, S.
Temperature Error Reduction of DPD Fluid by Using Partitioned Runge-Kutta Time Integration Scheme. *Fluids* **2019**, *4*, 156.
https://doi.org/10.3390/fluids4030156

**AMA Style**

Yamada T, Itoh S, Morinishi Y, Tamano S.
Temperature Error Reduction of DPD Fluid by Using Partitioned Runge-Kutta Time Integration Scheme. *Fluids*. 2019; 4(3):156.
https://doi.org/10.3390/fluids4030156

**Chicago/Turabian Style**

Yamada, Toru, Shugo Itoh, Yohei Morinishi, and Shinji Tamano.
2019. "Temperature Error Reduction of DPD Fluid by Using Partitioned Runge-Kutta Time Integration Scheme" *Fluids* 4, no. 3: 156.
https://doi.org/10.3390/fluids4030156