Unconstrained ETD Methods on the Diffuse-Interface Model with the Peng-Robinson Equation of State

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.