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 - <p style="text-align: justify;">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.</p>