A reduced order model using proper orthogonal decomposition and discrete empirical interpolation method for drop formation in a liquid jet

We develop a reduced-order model (ROM) for early-time droplet formation in continuous inkjet atomization using a one-dimensional jet model. A full-order model (FOM) provides snapshot data approaching breakup, from which proper orthogonal decomposition (POD) bases are extracted. The nonlinear terms are approximated with the discrete empirical interpolation method (DEIM), and the resulting POD–DEIM ROM is evaluated by time-domain comparisons and eigenvalue-based stability diagnostics. For the training cases and an interpolated parameter set, the ROM reproduces the pre-breakup evolution and necking dynamics up to approximately first 99% of duration of original simulation, with typical relative L2 errors below 1% through most of this interval. Accuracy degrades as the neck radius collapses, and some trajectories diverge as the solution approaches the pinch-off. Sweeps that increase POD and DEIM ranks and oversample the nonlinear sensors improve agreement before pinch-off but do not eliminate the late-time instability, which remains a limitation of the present configuration. A key quantitative finding concerns conditional stability and efficiency in the pre-breakup regime. At the matched time step size 10−6, speed-ups are order unity because the one-dimensional FOM is already inexpensive per step. However, the POD projection filters stiff high-frequency modes and reduces the Jacobian spectral radius, increasing the stability margin: the FOM diverges at the time step of 10−5, whereas selected POD–DEIM configurations remain stable through the pre-breakup window. This permits a tenfold larger time step size and an optional order-of-magnitude reduction in runtime for early-time prediction. Extending accuracy and stability through pinch-off, for example with residual-minimizing formulations such as LSPG, is deferred to future work.