An efficient spectral element phase-field method for incompressible multiphase flow simulation

This study develops a graphics processing unit (GPU)-accelerated spectral element phase-field method for efficient and accurate simulation of incompressible two-phase flows. The objective is to reduce the cost of the repeated elliptic subproblems that dominate Navier–Stokes–Cahn–Hilliard simulations while retaining high-order accuracy and robust interface resolution. The governing system is advanced with a decoupled time-integration strategy in which the phase-field, pressure, and velocity subproblems are solved sequentially. The dominant elliptic operators are reformulated as constant-coefficient Poisson and Helmholtz equations, so the corresponding spectral element operators can be assembled, diagonalized, and reused throughout the computation. A tensor-product GPU solver based on one-dimensional eigenvalue decompositions then performs multidimensional elliptic solves through separable transforms and pointwise operations rather than repeated construction of variable-coefficient matrices. The method is assessed through central processing unit (CPU)–GPU performance comparisons, manufactured-solution convergence tests, Laplace-law verification, bubble coalescence, rising-bubble benchmarks, and Rayleigh–Taylor instability. The results confirm high-order spatial accuracy and efficiency of the present method. The time cost for the two- and three-dimensional rising-bubble case are separately 2.9 and 43.0 min for 15 000 time steps. The GPU solver reaches a peak speedup of 7.30 times and remains 3.66 times faster than the CPU reference for the largest tested problem with 1.24×108 degrees of freedom, while the maximum CPU–GPU difference stays at the round-off level. These results indicate that the proposed method provides an accurate and computationally efficient framework for large-scale phase-field simulations of incompressible multiphase flows.