An Efficient Boundary Integral Scheme for the MBO Threshold Dynamics Method via the NUFFT

The MBO threshold dynamics method consists of two steps. The first step solves a pure initial value problem of the heat equation with the initial data being the indicator function of some bounded domain. In the second step, the new sharp interface is generated via thresholding either by some prescribed solution value or by volume preserving. We propose an efficient boundary integral scheme for simulating the threshold dynamics via the nonuniform fast Fourier transform (NUFFT). The first step is carried out by evaluating a boundary integral via the NUFFT, and the second step is performed applying a root-finding algorithm along the normal directions of a discrete set of points at the interface. Unlike most existing methods where volume discretization is needed for the whole computational domain, our scheme requires the discretization of physical space only in a small neighborhood of the interface and thus is meshfree. The algorithm is spectrally accurate in space for smooth interfaces and has \(O(N\log N)\) complexity, where N is the total number of discrete points near the interface when the time step \(\Delta t\) is not too small. The performance of the algorithm is illustrated via several numerical examples in both two and three dimensions.