  This paper proposes a Cartesian grid-based boundary integral method for
efficiently and stably solving two representative moving interface problems,
the Hele-Shaw flow and the Stefan problem. Elliptic and parabolic partial
differential equations (PDEs) are reformulated into boundary integral equations
and are then solved with the matrix-free generalized minimal residual (GMRES)
method. The evaluation of boundary integrals is performed by solving equivalent
and simple interface problems with finite difference methods, allowing the use
of fast PDE solvers, such as fast Fourier transform (FFT) and geometric
multigrid methods. The interface curve is evolved utilizing the $\theta-L$
variables instead of the more commonly used $x-y$ variables. This choice
simplifies the preservation of mesh quality during the interface evolution. In
addition, the $\theta-L$ approach enables the design of efficient and stable
time-stepping schemes to remove the stiffness that arises from the curvature
term. Ample numerical examples, including simulations of complex viscous
fingering and dendritic solidification problems, are presented to showcase the
capability of the proposed method to handle challenging moving interface
problems.

Numerical Analysis;Numerical Analysis;Computational Physics