A comparative study of two different mixed Eulerian-Lagrangian methods is presented. Representative numerical simulations of oscillatory flare-slamming flows are given. Computations based on these two different numerical schemes, i.e., a desingularized method using Rankine ring sources and a source-doublet panel method (e.g., USAERO/FSP©), are compared with experiments. Fourier coefficients of the simulated time histories and experimentally measured forces are given for detailed error comparisons. The numerical simulations demonstrate the ranges of applicability of these two methods. Both are shown to be efficient and robust time-stepping schemes for the fully nonlinear free-surface problem studied here.