A semi-implicit, second-order time-integration scheme for surface tension modeling in two-dimensional capillary-dominated two-phase flows

We present a second-order semi-implicit time integration scheme for modeling surface tension in capillary-dominated two-phase flows, implemented within a two-dimensional XFEM/level set framework. Traditional explicit methods are constrained by the capillary time-step, requiring prohibitively small increments and leading to long simulations and error accumulation. Semi-implicit strategies relax this limitation by introducing a Laplace-Beltrami operator that acts as a numerical interface viscosity, damping high-frequency interfacial modes and thereby improving stability. While this stabilizing dissipation enables larger time-steps and suppresses spurious currents, it also introduces artificial damping that may distort interfacial dynamics. Our proposed BDF2 formulation reduces this dissipation by a factor of two-thirds compared with the classical first-order variant, thus enhancing fidelity without sacrificing stability. The method is validated on three two-dimensional benchmarks: a static bubble, a rising bubble, and an oscillating bubble demonstrating robust convergence to theoretical and reference solutions. Results confirm that the second-order semi-implicit scheme achieves genuine second-order temporal accuracy with improved efficiency, making it suitable for high-fidelity simulations of capillary-driven two-phase flows.