A B S T R A C TA robust numerical methodology to predict equilibrium interfaces over arbitrary solid surfaces is developed. The kernel of the proposed method is the distance regularized level set equations (DRLSE) with techniques to incorporate the no-penetration and mass-conservation constraints. In this framework, we avoid reinitialization typically used in traditional level set methods. This allows for a more efficient algorithm since only one advection equation is solved, and avoids numerical error associated with the re-distancing step. A novel surface tension distribution, based on harmonic mean, is prescribed such that the zero level set has the correct the liquid-solid surface tension value. This leads to a more accurate triple contact point location. The method uses second-order central difference schemes which facilitates easy parallel implementation, and is validated by comparing to traditional level set methods for canonical problems. The application of the method, in the context of Gibbs free energy minimization, to obtain liquid-air interfaces is validated against existing analytical solutions. The capability of our current methodology to predict equilibrium shapes over both structured and realistic rough surfaces is demonstrated.

A novel technique based on the Full Orthogonalization Arnoldi (FOA) is proposed to perform Dynamic Mode Decomposition (DMD) for a sequence of snapshots. A modification to FOA is presented for situations where the matrix A is unknown, but the set of vectorsThe modified FOA is the kernel for the proposed projected DMD algorithm termed, FOA based DMD. The proposed algorithm to compute DMD modes and eigenvalues i) does not require Singular Value Decomposition (SVD) for snapshot matrices X with κ 2 (X) ≪ 1/ǫ m , where κ 2 (X) is the 2-norm condition number of the snapshot matrix and ǫ m is the relative round-off error or machine epsilon, ii) has an optional rank truncation step motivated by round off error analysis for snapshot matrices X with κ 2 (X) ≈ 1/ǫ m , iii) requires only one snapshot at a time, thus making it a 'streaming' method even with the optional rank truncation step, iv) consumes less memory and requires less floating point operations to obtain the projected matrix than existing projected DMD methods and v) lends itself to easy parallelism as the main computational kernel involves only vector additions, dot products and matrix vector products. The new technique is therefore well-suited for DMD of large datasets on parallel computing platforms. We show both theoretically and using numerical examples that for FOA based DMD without rank truncation, the finite precision error in the computed projection of the linear mapping is O(ǫ m κ 2 (X)). The proposed method is also compared to existing projected DMD methods for computational cost, memory consumption and relative round off error. Error indicators are presented that are useful to decide when to stop acquiring new snapshots. The proposed method is applied to several examples of numerical simulations of fluid flow.

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