# On the Numerical Solution of Sparse Linear Systems Emerging in Finite Volume Discretizations of 2D Boussinesq-Type Models on Unstructured Grids

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. The Numerical Model

#### 3.1. The FV Approximation

#### 3.2. The Resulting Linear System for the Velocity Field Recovery

#### 3.3. System’s Matrix Properties

## 4. Iterative Methods, Preconditioning and Reordering

#### 4.1. Application of Iterative Methods

**Ax**=

**b**, and we use a “toy” problem in which the right hand side vector $\mathbf{b}$ is computed adding the columns of the matrix $\mathbf{A}$. The initial guess used for the iterative methods discussed next was the zero vector. From now on all test cases presented will solve this “toy” problem unless otherwise stated. We mention again here that the system’s matrix depends on the numbering and the geometrical quantities of the mesh nodes. So, and since it is not depending on the unknown quantities of the BT model we can construct it and store it only once in the beginning of our simulation.

#### 4.2. Application of Preconditioning Methods

**A**. This entails a decomposition of the form $\mathbf{A}=\mathbf{LU}-\mathbf{R}$ where

**L**and

**U**have the same nonzero structure as the lower and the upper part of

**A**, respectively and

**R**is the residual matrix of the factorization. This incomplete factorization is known as ILU(0) and often leads to a crude approximation which in turn may result to the need of many iterations to reach converge. To remedy this, several alternative incomplete factorizations have been developed by allowing more fill-ins in

**L**and

**U**. In practice, and as to find the

**L**and

**U**, the Gaussian elimination process is used and a level of fill-in is attributed to each element which is dropped or not according to this value [24]. In general, a more accurate ILU factorization requires fewer iterations to converge but of course the preprocessing cost to compute the factors is higher. Incomplete factorizations that rely on the level of fill are blind to numerical values because elements that are dropped depend only on the structure of the matrix. Some methods are available based on dropping elements in the Gaussian elimination process according to their magnitude rather than their location.

#### 4.2.1. The ILU(0) Preconditioner

#### 4.2.2. The ILU(k) Preconditioner

#### 4.2.3. ILUT($p,\tau $) Preconditioner:

**L**part of the row and the p-th largest elements in the

**U**part of the row in addition to the diagonal element, which is always kept. In this work we use always $p=300.$ The goal of this dropping step is to control the number of elements per row.

## 5. Reordering

**A**and to reduce excessive fill-in in the factorization of the involved operators [38]. In most cases, reordering techniques tend to affect the rate of convergence of preconditioned Krylov subspace methods [25]. These algorithms aim to minimize the bandwidth of the matrix $\mathbf{A}$. Two different approaches were used in this work namely, the Cuthill–McKee (CMK) and the Reverse Cuthill–McKee (RCM) permutations.

#### Spatial Accuracy and Efficiency

## 6. Conclusions

- BiCGSTAB and GMRES iterative methods give almost similar results for the resulting systems, with the BiCGSTAB to have been proven more robust in some cases and is the method of choice following from this work.
- The usage of preconditioning and/or reordering is mandatory as to achieve convergence for the different mesh types used.
- Using preconditioning and reordering we gained convergence for (all) systems in every water depth. Using only preconditioning we were able to solve efficiently systems that have a small condition number (usually derived from equilateral grids).
- Using a drop tolerance $\tau ={10}^{-5}$ (for ILU(k) and ILUT preconditioners): CPU time using ILUT is less than that of using ILU(k) in average water depths. The usage of ILU(k) maybe more expensive in time but results on an overall the same CPU time in any water depth for the same grid resolution for convergence.
- As to correct the limitation of ILUT we decreased the drop tolerance and we observed that for larger water depths both iterative methods converge, but of course with an additional time cost. Like before the CPU time is independent on the relative water depth on each matrix.
- The Reverse Cuthill–McKee (RCM) ordering was proven more efficient compared to the Cuthill–McKee (CMK) ordering. This is found to greatly improve the efficiency of the ILUT preconditioner, since it constrains the factorized matrix to lie within a much narrower bandwidth and hence the incomplete factorization is generally more accurate for a prespecified amount of storage.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Median-dual computational cell implemented in the FV scheme (

**left**) and the computational cell used for the gradient of the divergence in (2) (

**right**).

**Figure 2.**Representative grid types: Equilateral, Orthogonal I, Orthogonal II, Distorted (left to right).

**Figure 3.**Matrix sparsity patterns for the four different mesh types shown in Figure 2 for ${N}_{x}=15$ with ${N}_{z}$ the number of non-zero elements.

**Figure 4.**Eigenvalues of three matrices using the equilateral type of grid with $\frac{h}{{h}_{N}}=\frac{1}{0.058}$ (

**top left**), $\frac{1}{0.031}$ (

**top right**) and $\frac{100}{0.058}$ (

**bottom**).

**Figure 5.**Eigenvalues of three matrices using the Orthogonal I type of grid with $\frac{h}{{h}_{N}}=\frac{1}{0.0456}$ (

**top left**), $\frac{1}{0.0232}$ (

**top right**) and $\frac{100}{0.0456}$ (

**bottom**).

**Figure 6.**CPU time versus variable still water level to ${h}_{N}$ ratio for GMRES (solid line) and BiCGStab (dashed line).

**Figure 7.**CPU time versus $h/{h}_{N}$ for GMRES (solid line) and BiCGStab (dashed line) using the ILU(k) preconditioner with $k=300$.

**Figure 8.**CPU time versus $h/{h}_{N}$ for GMRES (solid line) and BiCGStab (dashed line) applying the ILUT preconditioner with $\tau ={10}^{-5}$.

**Figure 9.**CPU time versus $h/{h}_{N}$ for GMRES (solid line) and BiCGStab (dashed line) using the ILU(k) preconditioner and CMK reordering.

**Figure 10.**CPU time versus variable water depth for GMRES (solid line) and BiCGStab (dashed line) using ILUT preconditioner with threshold ${10}^{-5}$ (

**left**) and ${10}^{-10}$ (

**right**) and CMK reordering, for Equilateral type of grid (

**up**) and for Orthogonal I (

**down**).

**Figure 11.**CPU time versus variable water depth for GMRES (solid line) and BiCGStab (dashed line) using ILU(k) preconditioner and RCM reordering, for Equilateral type of grids (

**left**) and for Orthogonal I types (

**right**).

**Figure 12.**CPU time versus variable water depth for GMRES (solid line) and BiCGStab (dashed line) using ILUT preconditioner with threshold ${10}^{-5}$ (

**left**) and ${10}^{-10}$ (

**right**) and RCM reordering for Equilateral type of grids (

**up**) and Orthogonal I types (

**down**).

**Table 1.**Total ($2N\times 2N$) and non-zero elements (${N}_{z}$) for the matrices produced for ${L}_{x}=Ly=1$ m with different ${N}_{x}$.

${\mathit{N}}_{\mathit{x}}$ | Equilateral | Orthogonal I | Orthogonal II | Distorted |
---|---|---|---|---|

15 | 352,836 (7339) | 925,444 (12,905) | 262,144 (6430) | 352,836 (7715) |

30 | 4,857,616 (28,436) | 13,853,284 (50,806) | 3,694,084 (25,163) | 4,857,616 (29,746) |

60 | 7,4132,100 (118,293) | 214,388,164 (201,931) | 55,383,364 (99,258) | 74,132,100 (118,293) |

**Table 2.**The ${h}_{N}$ values for the matrices produced for ${L}_{x}={L}_{y}=1$ m with different ${N}_{x}$.

${\mathit{N}}_{\mathit{x}}$ | Equilateral | Orthogonal I | Orthogonal II | Distorted |
---|---|---|---|---|

15 | 0.058 | 0.0456 | 0.0625 | 0.058 |

30 | 0.0301 | 0.0232 | 0.0323 | 0.0301 |

60 | 0.0152 | 0.017 | 0.0164 | 0.0152 |

**Table 3.**CPU time for the solution of linear systems resulting from Orthogonal I type of grids using the ILU(k) preconditioner, with $k=300$, preconditioner.

$\mathit{h}/{\mathit{h}}_{\mathit{N}}$ | GMRES (s)/Iterations | BiCGStab (s)/Iterations |
---|---|---|

0.1/0.0456 | 1.588515759/2 | 1.588515759/3 |

1.0/0.0456 | 1.525697947/2 | 1.581536055/3 |

10/0.0456 | 1.580311775/3 | 1.591470003/3 |

0.1/0.0232 | 87.64499593/2 | 87.37235212/3 |

10/0.0232 | - | 87.47821903/3 |

${\mathit{N}}_{\mathit{x}}$ | Equilateral (s) | Orthogonal I (s) |
---|---|---|

15 | 0.014 | 0.012 |

30 | 0.19 | 0.15 |

60 | 2.62 | 2.24 |

120 | - | - |

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**MDPI and ACS Style**

Delis, A.I.; Kazolea, M.; Gaitani, M.
On the Numerical Solution of Sparse Linear Systems Emerging in Finite Volume Discretizations of 2D Boussinesq-Type Models on Unstructured Grids. *Water* **2022**, *14*, 3584.
https://doi.org/10.3390/w14213584

**AMA Style**

Delis AI, Kazolea M, Gaitani M.
On the Numerical Solution of Sparse Linear Systems Emerging in Finite Volume Discretizations of 2D Boussinesq-Type Models on Unstructured Grids. *Water*. 2022; 14(21):3584.
https://doi.org/10.3390/w14213584

**Chicago/Turabian Style**

Delis, Anargiros I., Maria Kazolea, and Maria Gaitani.
2022. "On the Numerical Solution of Sparse Linear Systems Emerging in Finite Volume Discretizations of 2D Boussinesq-Type Models on Unstructured Grids" *Water* 14, no. 21: 3584.
https://doi.org/10.3390/w14213584