Kernel-free Boundary Integral Methods for Allen-Cahn and Cahn-Hilliard Equations on Irregular Domains

A unified kernel-free boundary integral (KFBI) framework is proposed for the Allen-Cahn and Cahn Hilliard equations with homogeneous no-flux boundary conditions on two- and three-dimensional irregular domains. With a stabilized first-order implici-explicit (IMEX) discretization, the Allen-Cahn update reduces to a Neumann modified Helmholtz problem. An auxiliary-variable reformulation reduces the Cahn-Hilliard update to subproblems of the same type without evaluating the Laplacian of the nonlinear source. The Cahn-Hilliard update requires two sequential solves for real shifts, while complex-conjugate shifts allow the real-valued solution to be reconstructed from a single complex solve. All subproblems are handled by the same KFBI solver that indirectly evaluates potentials through equivalent interface problems. To correct discrete mass defects in the Cahn-Hilliard update, a geometry-weighted discrete $Lš$ mass projection is constructed tailored to the KFBI treatment of irregular boundaries. Exterior overlap weights are redistributed to physical nodes, excluding auxiliary KFBI extension values from the mass calculation. Numerical experiments demonstrate second-order spatial convergence, first order convergence of paired temporal-error indicators, and preservation of the prescribed discrete mass. The reported source-free simulations exhibit energy decay, with cumulative projection-induced energy perturbations remaining small relative to the observed dissipation.