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Volume 35, Issue 4
Unconstrained ETD Methods on the Diffuse-Interface Model with the Peng-Robinson Equation of State

Menghuo Chen, Yuanqing Wu, Xiaoyu Feng & Shuyu Sun

Commun. Comput. Phys., 35 (2024), pp. 973-1002.

Published online: 2024-05

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  • Abstract

In this study, we apply first-order exponential time differencing (ETD) methods to solve benchmark problems for the diffuse-interface model using the Peng-Robinson equation of state. We demonstrate the unconditional stability of the proposed algorithm within the ETD framework. Additionally, we analyzed the complexity of the algorithm, revealing that computations like matrix multiplications and inversions in each time step exhibit complexity strictly less than $\mathcal{O}(n^2),$ where $n$ represents the number of variables or grid points. The main objective was to develop an algorithm with enhanced performance and robustness. To achieve this, we avoid iterative solutions (such as matrix inversion) in each time step, as they are sensitive to matrix properties. Instead, we adopted a hierarchical matrix ($\mathcal{H}$-matrix) approximation for the matrix inverse and matrix exponential used in each time step. By leveraging hierarchical matrices with a rank $k ≪ n,$ we achieve a complexity of $O(kn{\rm log}(n))$ for their product with an $n$-vector, which outperforms the traditional $\mathcal{O}(n^2)$ complexity. Overall, our focus is on creating an unconditionally stable algorithm with improved computational efficiency and reliability.

  • AMS Subject Headings

65F60, 65M22, 68Q25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-35-973, author = {Chen , MenghuoWu , YuanqingFeng , Xiaoyu and Sun , Shuyu}, title = {Unconstrained ETD Methods on the Diffuse-Interface Model with the Peng-Robinson Equation of State}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {4}, pages = {973--1002}, abstract = {

In this study, we apply first-order exponential time differencing (ETD) methods to solve benchmark problems for the diffuse-interface model using the Peng-Robinson equation of state. We demonstrate the unconditional stability of the proposed algorithm within the ETD framework. Additionally, we analyzed the complexity of the algorithm, revealing that computations like matrix multiplications and inversions in each time step exhibit complexity strictly less than $\mathcal{O}(n^2),$ where $n$ represents the number of variables or grid points. The main objective was to develop an algorithm with enhanced performance and robustness. To achieve this, we avoid iterative solutions (such as matrix inversion) in each time step, as they are sensitive to matrix properties. Instead, we adopted a hierarchical matrix ($\mathcal{H}$-matrix) approximation for the matrix inverse and matrix exponential used in each time step. By leveraging hierarchical matrices with a rank $k ≪ n,$ we achieve a complexity of $O(kn{\rm log}(n))$ for their product with an $n$-vector, which outperforms the traditional $\mathcal{O}(n^2)$ complexity. Overall, our focus is on creating an unconditionally stable algorithm with improved computational efficiency and reliability.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0256}, url = {http://global-sci.org/intro/article_detail/cicp/23091.html} }
TY - JOUR T1 - Unconstrained ETD Methods on the Diffuse-Interface Model with the Peng-Robinson Equation of State AU - Chen , Menghuo AU - Wu , Yuanqing AU - Feng , Xiaoyu AU - Sun , Shuyu JO - Communications in Computational Physics VL - 4 SP - 973 EP - 1002 PY - 2024 DA - 2024/05 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0256 UR - https://global-sci.org/intro/article_detail/cicp/23091.html KW - Diffuse-interface model, exponential time differencing method, hierarchical matrix. AB -

In this study, we apply first-order exponential time differencing (ETD) methods to solve benchmark problems for the diffuse-interface model using the Peng-Robinson equation of state. We demonstrate the unconditional stability of the proposed algorithm within the ETD framework. Additionally, we analyzed the complexity of the algorithm, revealing that computations like matrix multiplications and inversions in each time step exhibit complexity strictly less than $\mathcal{O}(n^2),$ where $n$ represents the number of variables or grid points. The main objective was to develop an algorithm with enhanced performance and robustness. To achieve this, we avoid iterative solutions (such as matrix inversion) in each time step, as they are sensitive to matrix properties. Instead, we adopted a hierarchical matrix ($\mathcal{H}$-matrix) approximation for the matrix inverse and matrix exponential used in each time step. By leveraging hierarchical matrices with a rank $k ≪ n,$ we achieve a complexity of $O(kn{\rm log}(n))$ for their product with an $n$-vector, which outperforms the traditional $\mathcal{O}(n^2)$ complexity. Overall, our focus is on creating an unconditionally stable algorithm with improved computational efficiency and reliability.

Chen , MenghuoWu , YuanqingFeng , Xiaoyu and Sun , Shuyu. (2024). Unconstrained ETD Methods on the Diffuse-Interface Model with the Peng-Robinson Equation of State. Communications in Computational Physics. 35 (4). 973-1002. doi:10.4208/cicp.OA-2023-0256
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