Flux-balanced Smoothed Particle Hydrodynamics with differential reproducing kernels for meshfree phase-field modeling of multiphase flow

We present a flux-balanced Smoothed Particle Hydrodynamics (FBSPH) formulation for meshfree phase-field (PF) modeling of multiphase flow, where pairwise anti-symmetric numerical fluxes ensure that the flux leaving a particle is exactly that entering its neighbor. Furthermore, Differential Reproducing Kernels (DRK) are used for constructing the kernel functions to a higher-order of consistency. FBSPH is applied to the conservative Allen–Cahn PF equations for immiscible two-phase and three-phase flows on both fixed and moving particles. In this framework, diffusive and advective fluxes are treated using a Symmetric Interior Penalty (SIP) formulation and a local Lax–Friedrichs Riemann solver, respectively. Benchmarks demonstrate that flux balancing conserves the PF order parameter close to machine precision on fixed particles. On moving particles, it significantly reduces the conservation error and inter-phase mass transfer relative to standard SPH with identical kernels. A comparison across kernel orders shows that conservation is governed by the flux treatment, while accuracy is governed by kernel consistency. The presented meshfree PF formulation is Galilean invariant, handles high density ratios, simplifies surface tension imposition in ternary systems, and suppresses stray particle formation. The mathematical formulation and numerical results for a comprehensive range of benchmarks are presented and discussed.