#
Implementation and Validation of Semi-Implicit WENO Schemes Using OpenFOAM^{®}

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}resulting in the access to a huge open-source community and the applicability to high-level programming. Several verification cases and applications of the scalar advection equation and the incompressible Navier-Stokes equations show the improved accuracy of the WENO approach due to a mapping of the stencil to a reference space without scaling effects. An efficiency analysis indicates an increased computational effort of high-order schemes in comparison to available high-resolution methods. However, the reconstruction time can be efficiently decreased when more processors are used.

## 1. Introduction

^{®}gained popularity in the recent past since it can handle unstructured grids with polyhedral cells and provides a high-level programming platform for applying new solvers easily. Unstructured meshes are currently in the focus of operators due to their increased versatility in the application CFD methods in complex geometries [1] and the reduced generation time. However, the irregular structure of the grid prevents most codes such as OpenFOAM

^{®}to reach more than second order of accuracy. Further, the commonly applied second-order methods such as total variation diminishing (TVD) schemes are derived for structured grids. This might cause unbounded solutions on unstructured grids dependent on the chosen gradient scheme.

^{®}. However, in his work, he restricted the usage to explicit convection terms in hyperbolic equations and provided less information about the implementation into the code itself.

^{®}using a similar approach to [3]. This provides high-order reconstructions on unstructured grids to a huge group of researchers and enables these developments to a broad scope of applications for the first time. The method is mainly based on the approach of Dumbser and Käser [16] and Tsoutsanis et al. [1]. Hence, the scheme operates in a reference space without scaling effects in order to prevent ill-conditioned matrices and improve the accuracy on irregular grids. All mesh types are handled automatically which is in accordance to OpenFOAM

^{®}. On these general grids, finite volume WENO schemes were usually applied to convection terms in an explicit manner. As an exceptional example, implicit discontinuous Galerkin methods based on hierarchical WENO reconstructions might be of interest to the reader [21]. The possibility to build a semi-implicit finite volume WENO convection scheme arises by including the reconstruction in OpenFOAM

^{®}due to existing code structures for such an implementation. It results in a more stable reconstruction for larger time steps and an utilization of high-order convection schemes in common semi-implicit solution algorithms such as SIMPLE or PISO (Pressure-Implicit with Splitting of Operators). However, the theoretical accuracy limitation of these algorithms can of course not be affected.

^{®}is described in detail. Afterwards, the derivations of semi-implicit WENO convection schemes (Section 3) and different gradient schemes (Section 4) are given. It follows Section 5 with implementation and application details. Section 6 is dedicated to the verification process of the schemes, whereas Section 7 shows several test cases including an efficiency comparison. A conclusion completes this article in Section 8.

## 2. Numerical Approach of WENO Reconstruction Methods

#### 2.1. Stencil Collection Algorithm

- All coordinates in the reference space are positive.
- The cell is not already a member of another stencil which may happen on Cartesian grids where the centre lies on the boundary of two adjunct sectors.
- The target stencil list contains no more than ${J}_{max}$ cells.

#### 2.2. Parallelisation

^{®}is based on a 0-halo approach which divides the domain into several non-overlapping regions and Message Passing Interface (MPI) to transmit the information between the inter-processor boundaries. This leads to at best second-order accurate solutions at such boundaries. In contrast, the stencils of a high-order (W)ENO scheme near processor patches need the geometrical and physical data from several layers of the neighbouring domain. Consequently, a n-halo approach with several overlapping sub-domains would be the proper choice. The implicit handling of the Navier-Stokes equations leads to algebraic systems of equations which are solved by linear, iterative solvers in OpenFOAM

^{®}. Since these solvers only work for 0-halo approaches, a n-halo approach is discarded. Instead, the solution is the virtual extension of the sub-domains by collecting halo cells from neighbouring processors in additional lists. Then, the field values of the halo cells are updated at the beginning of each runtime step which is computed on non-overlapping domains.

- For each sub-domain ${D}_{i}$, all cells from the stencils of target cells next to a processor boundary are gathered in a list of halo cells together with the information of the target processor. Beyond, the stencils of these cells are marked as possible acceptors for halo cells from other sub-domains. In Figure 3a, acceptor cells of the sub-domain ${D}_{1}$ are coloured green while its halo cells from sub-domain ${D}_{2}$ are coloured red and vice versa the green cells are the halo cells from sub-domain ${D}_{1}$ for the red acceptor cells of ${D}_{2}$.
- The lists of halo cells are further prepared by assigning them a new ID and additionally, storing their cell centre coordinates and the coordinates of the triangles from the triangulation of the cell’s boundaries. Afterwards, the lists are transmitted to the appropriate target processor using MPI.
- The required halo cells for each marked central stencil ${S}_{c,i}$ are determined by a geometrical selection due to missing face neighbour information beyond processor boundaries. For this purpose, a sphere is spanned around the target cell ${V}_{i}^{\prime}$ of ${S}_{c,i}$ with the distance from the centre of ${V}_{i}^{\prime}$ to the outermost cell centre in the local stencil as the radius. All halo cells whose cell centres are located within this sphere are added to a new global stencil. In Figure 3b, this geometrical selection results in the yellow coloured global stencil for the blue framed stencil in Figure 3a.
- The final stencils are attained from sorting the global stencils by distance and pick the nearest ${J}_{max}$ cells. In Figure 3b, the new stencil is framed in blue.

## 3. Derivation of Semi-Implicit WENO-based Convection Schemes

^{®}(The presented interpolations are, however, limited to the surface interpolation class of OpenFOAM

^{®}which stands in contrast to the point interpolation class).

^{®}is limited to second order of accuracy, the basis classes multiply the results of the interpolation by the face areas. Therefore, we neglect the surface area in (28) in the implementation in order stay consistent). The remaining surface integrals in (28) are solution independent and can be precomputed. Under consideration of (5), they are expressed as

^{®}such as SIMPLE or PISO. Unfortunately, as it is shown in [31], WENO schemes are not strictly bounded. The explicit correction term can, therefore, be unbounded and still influence the solution’s physical reliability.

^{®}. Hence, Equation (39) is evaluated as

## 4. Derivation of a WENO Gradient Scheme

^{®}offers a least-squares-based gradient scheme whose stencils are however limited to the first neighbours. On the contrary, the presented scheme takes a larger stencil into account and avoids spurious oscillations at the same time. Alternatively, the WENO weighting could also be skipped in order to get a high-order version of the existing least-squares method.

^{®}due to its explicit treatment in any case.

## 5. Implementation of the WENO Schemes in OpenFOAM^{®}

^{®}are provided. The main accomplishment of this step is embedding the new methodology in the existing code structure such that all schemes are applicable in the same way as the existing low-order methods.

#### 5.1. Preprocessing

- Generation of one large central stencil list for each control volume ${V}_{i}^{\prime}$ in the transformed space and sorting it by distance as described in Section 2.1.
- Calculation and storing of all volume integrals of the basis functions of ${V}_{i}^{\prime}$ using triangularization of the faces and a Gaussian quadrature rule of appropriate order.
- If several processors are involved, halo cells are collected. Afterwards, appropriate cell coordinates and triangulated face coordinates are transmitted and the global stencils are gathered using the procedure of Section 2.2.
- Generation of the sectoral stencils and the final central stencil in accordance with the algorithm described in Section 2.1.
- Determination of the reconstruction matrix for each sectoral stencil of ${V}_{i}^{\prime}$ (see (21)). For this purpose, volume integrals of the basis functions for the cells in the stencil are calculated in the space of ${V}_{i}^{\prime}$. Finally, the pseudoinverse ${\mathcal{A}}^{+}$ is computed using SVD.
- Calculation of the oscillation indicator matrix $\mathcal{B}$ for each ${V}_{i}^{\prime}$ using (14).

#### 5.2. Runtime

- Collection and transmission of the field values of the halo cells in case of parallel computing.
- For each stencil of ${V}_{i}^{\prime}$, generation of the vector b as the right hand side of (19) using $\overline{\Phi}$. The degrees of freedom ${a}_{k}^{\left(m\right)}$ are then computed directly from a matrix vector product using ${\mathcal{A}}^{+}$.
- Insertion of the coefficients in (10) and evaluation of the smoothness indicator afterwards.

## 6. Verification

#### 6.1. Accuracy of Reconstruction for Smooth Functions

#### 6.2. Numerical Convergence Study of the Advection Equation

## 7. Applications

#### 7.1. Application to the Gradient Calculation

^{®}’s standard method, selectable as Gauss linear, are applied to the harmonic Function (51) of Section 6.1.

#### 7.2. Application to the Advection Equation

^{®}.

#### 7.3. Three-Dimensional Breaking of a Dam

^{®}’s interFoam solver, which is the standard solver for incompressible two-phase flows using RANS approach, is taken as a reference. It is based on a volume of fluid method with a special compression term for avoiding smearing of interfaces. In order to avoid unphysical fluid properties from improper advection and compression, the limiting strategy MULES is introduced. It has the objective to ensure boundedness of the solution of the volume fraction in each time step. Interpreting MULES as a flux-corrected method, it is obvious that a low-order, as well as a high-order advection flux has to be provided. Here, the interface model can benefit from the developed high-order WENOUpwindFit scheme. The nominal order of accuracy is, however, reduced by applying this limiting strategy. It can be shown that interFoam may lead to oscillatory interfaces in situations where the compression acts in wrong directions. In this connection, the authors added a relaxation equation with a novel diffusion coefficient based on the idea of Rusche [44]. The derivation of the resulting clsMULESFoam solver can be found in [27].

^{®}’s methods predict often over- and undershoots while the high-order convection results approximate the experimental distributions with a higher accuracy. This is, in particular, noticeable for the second wave front between t = 4 s and t = 5 s at ${H}_{2},\dots ,{H}_{4}$. At the last measuring point, both solvers have difficulties to predict the experimental distribution in the time interval t = [1.2 s, 2.8 s]. It should be, however, noticed that at this time the water behind the obstacle is strongly fluctuating which complicates both the experimental and numerical determination of the actual columns height.

#### 7.4. Performance Comparison

## 8. Conclusions

^{®}which simplifies further working with the code in high-level programming and allows the development of a wider range of high-order convection and gradient schemes. Further, the reconstruction process can be taken as the basis of high-order Riemann solver which extends the possible scope of OpenFOAM

^{®}significantly.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Layers of cells added at each iteration: in yellow, considered cell; in purple, initial neighbours; in red, the cells added at the first iteration; in green, the cells added at the second iteration—two-dimensional example.

**Figure 2.**Definition of the three sectoral stencils for a triangular cell: in yellow, considered cell–two-dimensional example.

**Figure 3.**Stencil collection algorithm at processor boundaries. (

**a**) Local stencil; (

**b**) global stencil. In green, cells of processor ${D}_{1}$; in red, cells of processor ${D}_{2}$; in yellow, final stencil.

**Figure 5.**Two-dimensional meshes for the verification process. (

**a**) 2500 hexahedra; (

**b**) 6172 triangular prisms.

**Figure 6.**Three-dimensional meshes for the verification process. (

**a**) 8000 hexahedra; (

**b**) 7982 tetrahedra.

**Figure 7.**Initial data for the reconstruction of the smooth functions. (

**a**) 2D function; (

**b**) 3D function.

**Figure 8.**${\mathcal{L}}_{2}$-norm of the error for the smooth, two-dimensional function. (

**a**) 2500 hexahedra; (

**b**) 6172 triangles.

**Figure 9.**${\mathcal{L}}_{2}$-norm of the error for the smooth, three-dimensional function. (

**a**) Hexahedra; (

**b**) tetrahedra.

**Figure 10.**Results of the gradient calculation in the principal directions along the diagonal d. (

**a**) 5800 hexahedra; (

**b**) 6048 tetrahedra.

**Figure 14.**Slice of a three-dimensional step-profile at z = 0.5 m on a mesh with 7982 tetrahedra. (

**a**) TVD-vanLeer; (

**b**) limited WENOUpwindFit 3.

**Figure 16.**Results of the limited and unlimited third-order WENO scheme for Zalesak’s disk on $128\times 128$ squares. (

**a**) Contour plot of $\Phi =0.5$; (

**b**) slice of the disk at y = 0.6 m.

**Figure 17.**Results of convection schemes for Zalesak’s disk on $128\times 128$ squares. (

**a**) Contour plot of $\Phi =0.5$; (

**b**) slice of the disk at y = 0.6 m.

**Figure 18.**Results of convection schemes for Zalesak’s disk on $256\times 256$ squares. (

**a**) Contour plot of $\Phi =0.5$; (

**b**) slice of the disk at y = 0.6 m.

**Figure 19.**Results of convection schemes for Zalesak’s disk on $128\times 128$ triangles. (

**a**) Contour plot of $\Phi =0.5$; (

**b**) slice of the disk at y = 0.6 m.

**Figure 20.**Results of convection schemes for Zalesak’s disk on $256\times 256$ triangles. (

**a**) Contour plot of $\Phi =0.5$; (

**b**) slice of the disk at y = 0.6 m.

**Figure 23.**Measured water column heights over time. (

**a**) ${H}_{1}$ over time; (

**b**) ${H}_{2}$ over time; (

**c**) ${H}_{3}$ over time; (

**d**) ${H}_{4}$ over time.

**Figure 24.**Time for one discretisation step related to TVD-vanLeer on 32 processors using different schemes and number of processors. All computations are executed using the interFoam solver.

**Figure 25.**Percentage of single sub-steps of a WENO reconstruction. Explanation for the legend: Get lists—receive necessary lists from WENOBase, Swap lists—interprocessor communication and transmission of data of halo cells, Calculating ${a}_{k}$—matrix vector multiplication for obtaining coefficients of sub-stencils, Calculating ${\tilde{a}}_{k}$—weighting ${a}_{k}$ and summing up.

**Table 1.**Order of accuracy from the numerical convergence study for the smooth, two-dimensional function on hexahedral ${\alpha}_{hex}$ and triangular ${\alpha}_{t}$ grids using ${\mathcal{L}}_{2}$.

Polynomial Order r | ${\mathit{\alpha}}_{\mathit{hex}}$ | ${\mathit{\alpha}}_{\mathit{t}}$ |
---|---|---|

1 | $2.1$ | $1.7$ |

2 | $3.1$ | $3.2$ |

3 | $4.7$ | $4.7$ |

4 | $6.2$ | $5.1$ |

**Table 2.**Order of accuracy from the numerical convergence study for the smooth, three-dimensional function on hexahedral ${\alpha}_{hex}$ and tetrahedral ${\alpha}_{tet}$ grids using ${\mathcal{L}}_{2}$.

Polynomial Order r | ${\mathit{\alpha}}_{\mathit{hex}}$ | ${\mathit{\alpha}}_{\mathit{tet}}$ |
---|---|---|

1 | $2.0$ | $1.9$ |

2 | $3.1$ | $3.3$ |

3 | $4.1$ | $4.2$ |

**Table 3.**Error and convergence of WENO convection schemes for the advection equation on hexahedral meshes.

Convection Scheme | Number of Cells | ${\mathcal{L}}_{2}$-Norm of Error | $\mathit{\alpha}$ |
---|---|---|---|

TVD-vanLeer | 4096 | $0.75$ | - |

27,000 | $0.41$ | $0.96$ | |

64,000 | $0.25$ | $1.62$ | |

125,000 | $0.17$ | $1.75$ | |

WENO1 | 4096 | $0.88$ | - |

27,000 | $0.62$ | $0.57$ | |

64,000 | $0.34$ | $2.07$ | |

125,000 | $0.21$ | $1.98$ | |

WENO2 | 4096 | $0.7$ | - |

27,000 | $0.25$ | $1.67$ | |

64,000 | $0.10$ | $3.13$ | |

125,000 | $0.05$ | $2.95$ | |

WENO3 | 4096 | $0.66$ | - |

$\mathrm{27,000}$ | $0.12$ | $2.66$ | |

$\mathrm{64,000}$ | $0.05$ | $3.35$ | |

$\mathrm{125,000}$ | $0.02$ | $3.86$ |

**Table 4.**Error and convergence of WENO convection schemes for the advection equation on tetrahedral meshes.

Convection Scheme | Number of Cells | ${\mathcal{L}}_{2}$-Norm of Error | $\mathit{\alpha}$ |
---|---|---|---|

TVD-vanLeer | 4188 | $0.83$ | - |

30,255 | $0.64$ | $0.39$ | |

58,456 | $0.49$ | $1.2$ | |

133,026 | $0.31$ | $1.75$ | |

WENO1 | 4188 | $0.87$ | - |

30,255 | $0.56$ | $0.65$ | |

58,456 | $0.29$ | $2.94$ | |

133,026 | $0.16$ | $2.26$ | |

WENO2 | 4188 | $0.76$ | - |

30,255 | $0.35$ | $1.17$ | |

58,456 | $0.21$ | $2.22$ | |

133,026 | $0.09$ | $3.1$ | |

WENO3 | 4188 | $0.53$ | - |

30,255 | $0.04$ | $3.95$ | |

58,456 | $0.02$ | $4.38$ | |

133,026 | $0.006$ | $3.34$ |

Solver | ${\mathit{H}}_{1}$ | ${\mathit{H}}_{2}$ | ${\mathit{H}}_{3}$ | ${\mathit{H}}_{4}$ |
---|---|---|---|---|

clsMULESFoam | $0.110$ | $0.057$ | $0.042$ | $0.071$ |

interFoam | $0.153$ | $0.107$ | $0.075$ | $0.079$ |

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**MDPI and ACS Style**

Martin, T.; Shevchuk, I.
Implementation and Validation of Semi-Implicit WENO Schemes Using OpenFOAM^{®}. *Computation* **2018**, *6*, 6.
https://doi.org/10.3390/computation6010006

**AMA Style**

Martin T, Shevchuk I.
Implementation and Validation of Semi-Implicit WENO Schemes Using OpenFOAM^{®}. *Computation*. 2018; 6(1):6.
https://doi.org/10.3390/computation6010006

**Chicago/Turabian Style**

Martin, Tobias, and Ivan Shevchuk.
2018. "Implementation and Validation of Semi-Implicit WENO Schemes Using OpenFOAM^{®}" *Computation* 6, no. 1: 6.
https://doi.org/10.3390/computation6010006