# A Study on the Numerical Performances of Diffuse Interface Methods for Simulation of Melting and Their Practical Consequences

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## Abstract

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## 1. Introduction

`Matlab`optimized code with an enthalpy or source term method against a non-optimized in-house

`C++`code for a heat capacity method. Finally, when using a coarse grid, Swaminathan and Voller showed that the source-term method was rather slow for isothermal phase change but also that an optimal formulation could limit these caveats [15].

## 2. Problem Statement

#### 2.1. Flow Modeling

#### 2.2. Thermal Modeling

**Happarent**solver [22,23] corresponds to the usual energy equation, written either for AE (Equation (4a)) or for SE (Equation (4b)):$$\begin{array}{c}\hfill {\displaystyle \rho \left(\right)open="("\; close=")">{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}h}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}h}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\lambda \overrightarrow{\nabla}T\hfill \end{array}$$$$\begin{array}{c}\hfill {\displaystyle \rho \left(\right)open="("\; close=")">{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}h}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+c\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}T}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\lambda \overrightarrow{\nabla}T\hfill \end{array}$$**Hsource**solver [24] relies on writing Fourier’s law in terms of the enthalpy, which leads to the following variants for AE (Equation (5a)) and SE (Equation (5b)):$$\begin{array}{c}\hfill {\displaystyle \rho \left(\right)open="("\; close=")">{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}h}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}h}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{\displaystyle \frac{\lambda}{c}}\overrightarrow{\nabla}h-{\displaystyle \frac{\lambda}{c}}L\Delta {\alpha}_{l}\hfill \end{array}$$$$\begin{array}{c}\hfill {\displaystyle \rho \left(\right)open="("\; close=")">{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}h}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+c\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}T}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{\displaystyle \frac{\lambda}{c}}\overrightarrow{\nabla}h-{\displaystyle \frac{\lambda}{c}}L\Delta {\alpha}_{l}\hfill \end{array}$$**Tapparentlinear**solver uses a modified specific capacity to take into account the latent effects. AE (Equation (6a)) and SE (Equation (6b)) cases lead to:$$\begin{array}{c}\hfill {\displaystyle \rho \phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{c}_{\mathrm{app}}{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}T}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}h}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\lambda \overrightarrow{\nabla}T\hfill \end{array}$$$$\begin{array}{c}\hfill {\displaystyle \rho \left(\right)open="("\; close=")">{c}_{\mathrm{app}}{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}T}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+c\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}T}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\lambda \overrightarrow{\nabla}T\hfill \end{array}$$In Equation (6a), the enthalpy is used in the convective term as a formulation with the temperature and the apparent heat capacity leading to an unstable behavior [6]. Among the variable possibilities, the expression for the apparent heat capacity is the one of Morgan et al. [25]:$$c}_{\mathrm{app}}={\displaystyle \frac{{h}^{n}-{h}^{n-1}}{{T}^{n}-{T}^{n-1}}$$**Tsource**solver [27] is based on the splitting of the enthalpy into its sensible and latent parts and becomes Equations (8a) and (8b) for AE and SE respectively.$$\begin{array}{c}\hfill {\displaystyle \rho c\left(\right)open="("\; close=")">{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}T}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}T}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\lambda \overrightarrow{\nabla}T-\rho L\left(\right)open="("\; close=")">{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}{\alpha}_{l}}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\alpha}_{l}\hfill \end{array}$$$$\begin{array}{c}\hfill {\displaystyle \rho c\left(\right)open="("\; close=")">{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}T}{\partial \phantom{\rule{0.166667em}{0ex}}t}}+\overrightarrow{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}T}\\ =\overrightarrow{\nabla}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\lambda \overrightarrow{\nabla}T-\rho L{\displaystyle \frac{\partial \phantom{\rule{0.166667em}{0ex}}{\alpha}_{l}}{\partial \phantom{\rule{0.166667em}{0ex}}t}}\hfill \end{array}$$

#### 2.3. Numerical Implementation

#### 2.4. Threshold and Evaluation

## 3. Test Case

## 4. Results

#### 4.1. Summary of the Liquid Fraction Evaluation

#### 4.2. Total Iterations

- On average, SE needs fewer iterations than AE except for Happarent where AE needs slightly fewer than SE.
- The differences between the minimum and the maximum number of iterations within one solver, but for different parameter combinations are between one and two orders of magnitude.
- For SE, the ranking is in ascending order: Tlinear, Hsource, Tsource, Happarent, Tapparentlinear.
- For AE, the ranking is in ascending order: Tlinear, Happarent, Hsource, Tsource, Tapparentlinear.

#### 4.3. Iterations per Time Step

#### 4.4. Computational Time

#### 4.5. Computational Time per Iteration

#### 4.6. Underrelaxation

## 5. Discussion

#### 5.1. General Results

#### 5.2. Solver Specific Analysis

**Happarent**was resilient for AE as well as SE for a broad range of parameters. Although it needed the strongest underrelaxation, it required one of the fewest iterations for AE and was in midfield of needed iterations for SE. The SE formulation of Happarent is the only solver not relying on the solving of a matrix equation system for the energy equation—resulting in the shortest mean simulation time per iteration. The implementation of Happarent is very simple so that it is the easiest solver to program.**Hsource**was the most resilient solver for the tested conditions; for both formulations, no simulations gave results outside the chosen thresholds of a final liquid fraction from 60% to 70%. It is also the solver that was the least dependent on underrelaxation—only the AE formulation needs underrelaxation in very few cases. In terms of simulation time and required iterations, Hsource is within the average of the resilient solvers. The implementation of Hsource is moderately complex.**Tsource**has proven to be resilient for a wide range of parameters and only in very few cases underrelaxation was needed. Tsource is within the average of the resilient solvers concerning the simulation time and required iterations. In addition, the implementation of Tsource is moderately complex.**Tlinear**AE was resilient and Tlinear SE was the most resilient together with Hsource—no simulation gave results that were outside the chosen threshold of a final liquid fraction from 60% to 70%. It should be mentioned that a parallel study [30] indicates that an additional linearization of the convection term in the case of the AE formulation would increase its resilience. Underrelaxation was necessary in roughly 50% of the cases. Tlinear SE needed clearly the fewest iterations of all solvers followed by Tlinear AE. The fact that Tlinear needs fewer iterations than other solvers confirms results in the literature [26]. On the other hand, the calculation time for one iteration was long compared to the other solvers. The implementation of Tlinear is the most complex one of all solvers tested.**Tapparentlinear**in its present implementation (using the approach of Morgan et al. [25] with additional iterative correction of the apparent heat capacity) performed by far the worst and cannot be recommended. It was found that the apparent heat capacity correction can be trapped in between several points for this formulation [14]. However, it should be mentioned that many different apparent heat capacity methods exist and it is conceivable that others work better.

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

Latin letters | |

$\mathcal{A}$ | porosity function ($\mathrm{kg}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}{\phantom{\rule{0.277778em}{0ex}}}^{-3}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

c | specific capacity ($\mathrm{J}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}{\phantom{\rule{0.277778em}{0ex}}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

C | Darcy constant ($\mathrm{kg}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}{\phantom{\rule{0.277778em}{0ex}}}^{-3}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

g | gravity acceleration ($\mathrm{m}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}{\phantom{\rule{0.277778em}{0ex}}}^{-2}$) |

h | specific enthalpy ($\mathrm{J}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

L | latent heat ($\mathrm{kJ}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

P | pressure ($\mathrm{Pa}\phantom{\rule{0.277778em}{0ex}}$}) |

q | small Darcy constant (–) |

T | temperature ($\mathrm{K}\phantom{\rule{0.277778em}{0ex}}$}) |

$\overrightarrow{V}$ | velocity ($\mathrm{m}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

t | time (s) |

Greek letters | |

$\alpha $ | volume fraction (–) |

$\beta $ | volumetric coefficient of thermal expansion ($\mathrm{K}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

$\u03f5$ | tolerance (–) |

$\lambda $ | thermal conductivity ($\mathrm{W}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}{\phantom{\rule{0.277778em}{0ex}}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

$\rho $ | density ($\mathrm{kg}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}{\phantom{\rule{0.277778em}{0ex}}}^{-3}$) |

$\mu $ | dynamic viscosity ($\mathrm{kg}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}{\phantom{\rule{0.277778em}{0ex}}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

$\nu $ | kinematic viscosity ($\mathrm{m}{\phantom{\rule{0.277778em}{0ex}}}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}{\phantom{\rule{0.277778em}{0ex}}}^{-1}$) |

Exponents and subscripts | |

app | apparent |

l | liquid |

liq | liquidus |

M | melting |

ref | reference |

s | solid |

sol | solidus |

Acronyms | |

AE | all enthalpy convection |

CFL | Courant–Friedrichs–Lewy |

DIM | diffuse interface method |

FGM | fixed grid method |

Happ | Happarent |

Hsou | Hsource |

SE | sensible enthalpy convection |

Tapp | Tapparentlinear |

Tlin | Tlinear |

Tsou | Tsource |

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**Figure 1.**Examples of the enthalpy formulation [14].

**Figure 5.**Mean number of iterations in function of the various parameters for the AE and the SE case (clear and striped, respectively).

**Figure 6.**Min, mean and max number of iterations per time step for the AE and the SE case (clear and striped respectively).

Property | Value | Unit |
---|---|---|

${\rho}_{s}$ | 6093 | kg m^{−3} |

${\rho}_{l}$ | 6093 | kg m^{−3} |

${c}_{s}$ | 381.5 | J K^{−1} kg^{−1} |

${c}_{l}$ | 381.5 | J K^{−1} kg^{−1} |

${\lambda}_{s}$ | 32 | W K^{−1} m^{−1} |

${\lambda}_{l}$ | 32 | W K^{−1} m^{−1} |

${T}_{M}$ | 29.78 | °C |

L | 80 160 | J kg^{−1} |

$\mu $ | $1.81\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ | kg m^{−1} s^{−1} |

$\nu $ | $0.29706\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-6}$ | m^{2} s^{−1} |

$\beta $ | $1.2\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | K^{−1} |

Variable | Remarks | Variation | Unit |
---|---|---|---|

$\Delta T$ | Tlinear, Tapparentlinear | ${10}^{-2},{10}^{-1},2\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-1},5\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-1},1,2$ | $\mathrm{K}\phantom{\rule{0.277778em}{0ex}}$ |

Tsource | ${10}^{-2},2\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-1},2$ | $\mathrm{K}\phantom{\rule{0.277778em}{0ex}}$ | |

Happarent, Hsource | $0,2\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-1},2$ | $\mathrm{K}\phantom{\rule{0.277778em}{0ex}}$ | |

$\Delta x$ | number of cells | 50 × 50, 100 × 100, 150 × 150, 200 × 200 | − |

$\Delta t$ | time step | 0.1, 1 | $\mathrm{s}\phantom{\rule{0.277778em}{0ex}}$ |

$\mathrm{CFL}$ | 1, 4, 8 | − | |

$\mathrm{tol}$ | tolerance for the energy equation | ${10}^{-6}$, ${10}^{-10}$ | − |

$\mathrm{form}$ | formulation of the convective term | AE, SE | − |

Parameter | AE | SE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Variation | Happ | Hsou | Tsou | Tlin | Tapp | Happ | Hsou | Tsou | Tlin | Tapp | |

$\Delta x$ | 50 × 50 | 1.000 | 1.000 | 1.000 | 1.000 | 0.306 | 1.000 | 1.000 | 1.000 | 1.000 | 0.597 |

100 × 100 | 1.000 | 1.000 | 1.000 | 1.000 | 0.375 | 1.000 | 1.000 | 1.000 | 1.000 | 0.569 | |

150 × 150 | 1.000 | 1.000 | 0.944 | 0.972 | 0.347 | 1.000 | 1.000 | 0.972 | 1.000 | 0.431 | |

200 × 200 | 0.778 | 1.000 | 0.861 | 0.944 | 0.181 | 0.778 | 1.000 | 0.917 | 1.000 | 0.236 | |

$\Delta t$ | 1 s | 0.889 | 1.000 | 0.931 | 0.972 | 0.264 | 0.889 | 1.000 | 0.944 | 1.000 | 0.403 |

0.1 s | 1.000 | 1.000 | 0.972 | 0.986 | 0.340 | 1.000 | 1.000 | 1.000 | 1.000 | 0.514 | |

$\u03f5$ | ${10}^{-6}$ | 0.944 | 1.000 | 0.917 | 0.958 | 0.257 | 0.944 | 1.000 | 0.944 | 1.000 | 0.396 |

${10}^{-10}$ | 0.944 | 1.000 | 0.986 | 1.000 | 0.347 | 0.944 | 1.000 | 1.000 | 1.000 | 0.521 | |

CFL | 1 | 1.000 | 1.000 | 1.000 | 1.000 | 0.438 | 1.000 | 1.000 | 1.000 | 1.000 | 0.542 |

4 | 0.917 | 1.000 | 1.000 | 0.969 | 0.250 | 0.917 | 1.000 | 1.000 | 1.000 | 0.427 | |

8 | 0.917 | 1.000 | 0.854 | 0.969 | 0.219 | 0.917 | 1.000 | 0.917 | 1.000 | 0.406 | |

$\Delta T$ | 2 K | 1.000 | 1.000 | 0.958 | 1.000 | 0.688 | 1.000 | 1.000 | 0.958 | 1.000 | 0.708 |

1 K | - | - | - | 1.000 | 0.521 | - | - | - | 1.000 | 0.458 | |

0.5 K | - | - | - | 1.000 | 0.521 | - | - | - | 1.000 | 0.604 | |

0.2 K | 0.917 | 1.000 | 0.958 | 1.000 | 0.083 | 0.917 | 1.000 | 0.979 | 1.000 | 0.646 | |

0.1 K | - | - | - | 1.000 | 0.000 | - | - | - | 1.000 | 0.333 | |

0 K | 0.917 | 1.000 | 0.938 | 0.875 | 0.000 | 0.917 | 1.000 | 0.979 | 1.000 | 0.000 |

Parameter | AE | SE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Variation | Happ | Hsou | Tsou | Tlin | Tapp | Happ | Hsou | Tsou | Tlin | Tapp | |

$\Delta x$ | 50 × 50 | 1.878 | 2.418 | 4.262 | 1.638 | 32.011 | 2.897 | 2.403 | 3.745 | 1.067 | 27.470 |

100 × 100 | 2.936 | 3.968 | 8.191 | 2.808 | 10.977 | 6.360 | 4.034 | 6.518 | 1.048 | 27.007 | |

150 × 150 | 2.558 | 6.532 | 12.141 | 3.527 | 9.497 | 12.228 | 6.005 | 9.509 | 1.016 | 23.352 | |

200 × 200 | 2.789 | 5.401 | 10.372 | 1.708 | 16.653 | 10.583 | 6.808 | 12.049 | 1.012 | 30.389 | |

$\Delta t$ | 1 s | 2.596 | 4.018 | 7.152 | 1.966 | 14.728 | 8.808 | 4.010 | 6.782 | 1.069 | 17.906 |

0.1 s | 2.484 | 5.141 | 10.137 | 2.921 | 19.174 | 7.040 | 5.615 | 8.909 | 1.002 | 34.894 | |

$\u03f5$ | ${10}^{-6}$ | 1.370 | 6.139 | 14.863 | 4.319 | 30.659 | 8.672 | 3.189 | 9.428 | 1.068 | 34.982 |

${10}^{-10}$ | 3.713 | 3.020 | 3.014 | 1.014 | 10.717 | 7.220 | 6.436 | 6.502 | 1.003 | 25.128 | |

CFL | 1 | 3.556 | 3.720 | 8.948 | 1.009 | 31.252 | 7.971 | 3.654 | 6.450 | 1.000 | 40.446 |

4 | 2.123 | 4.964 | 9.869 | 3.404 | 8.329 | 11.317 | 7.736 | 12.519 | 1.070 | 25.253 | |

8 | 1.945 | 5.055 | 7.290 | 2.998 | 8.696 | 4.550 | 3.048 | 4.826 | 1.038 | 18.398 | |

$\Delta T$ | 2 K | 2.726 | 2.816 | 5.908 | 2.713 | 11.819 | 6.834 | 3.497 | 6.380 | 1.150 | 6.775 |

1 K | - | - | - | 1.007 | 16.747 | - | - | - | 1.000 | 30.995 | |

0.5 K | - | - | - | 1.000 | 17.355 | - | - | - | 1.000 | 35.913 | |

0.2 K | 2.274 | 5.324 | 9.572 | 4.178 | 38.320 | 7.061 | 5.152 | 7.833 | 1.039 | 35.391 | |

0.1 K | - | - | - | 1.000 | NaN | - | - | - | 1.000 | 31.088 | |

0 K | 2.625 | 5.599 | 10.626 | 4.923 | NaN | 9.943 | 5.789 | 9.581 | 1.025 | NaN |

**Table 5.**Solver specific summary of the overall results including practical aspects. The characters indicate how the solvers perform in each area. The order is ++, +, o, - and --, where ++ is the most favorable rating.

AE | SE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Happ | Hsou | Tsou | Tlin | Tapp | Happ | Hsou | Tsou | Tlin | Tapp | |

resilience | + | ++ | + | + | -- | + | ++ | + | ++ | -- |

iterations | o | o | - | + | -- | o | + | o | ++ | -- |

time per iteration | - | o | o | - | o | + | o | o | - | o |

numerical implementation | + | o | o | - | o | + | o | o | - | o |

underrelaxation | -- | + | + | o | -- | -- | ++ | + | o | -- |

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**MDPI and ACS Style**

König-Haagen, A.; Franquet, E.; Faden, M.; Brüggemann, D.
A Study on the Numerical Performances of Diffuse Interface Methods for Simulation of Melting and Their Practical Consequences. *Energies* **2021**, *14*, 354.
https://doi.org/10.3390/en14020354

**AMA Style**

König-Haagen A, Franquet E, Faden M, Brüggemann D.
A Study on the Numerical Performances of Diffuse Interface Methods for Simulation of Melting and Their Practical Consequences. *Energies*. 2021; 14(2):354.
https://doi.org/10.3390/en14020354

**Chicago/Turabian Style**

König-Haagen, Andreas, Erwin Franquet, Moritz Faden, and Dieter Brüggemann.
2021. "A Study on the Numerical Performances of Diffuse Interface Methods for Simulation of Melting and Their Practical Consequences" *Energies* 14, no. 2: 354.
https://doi.org/10.3390/en14020354