Unconstrained ETD Methods on the Diffuse-Interface Model with the Peng-Robinson Equation of State (Communications in Computational Physics, 2024)
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 $n$ representsthe number of variables or grid points. The main objective was to develop an algorithm with enhanced performance and robustness. To achieve this, we avoid iterativesolutions (such as matrix inversion) in each time step, as they are sensitive to matrixproperties. Instead, we adopted a hierarchical matrix ($\mathcal{H}$-matrix) approximation forthe matrix inverse and matrix exponential used in each time step. By leveraging hierarchical matrices with a rank $k ≪ n,$ we achieve a complexity of $O(kn{\rm log}(n))$ fortheir 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 improvedcomputational efficiency and reliability.