On the numerical approximation of a phase-field volume reconstruction model: Linear and energy-stable leap-frog finite difference scheme

Three-dimensional (3D) volume reconstruction plays a vital role in industries like 3D printing, prosthetic devices, medical imaging, and computer vision. Our paper is aiming to engineer a stable and precise leap-frog scheme that guarantees desired accuracy for a phase-field model aimed at 3D volume reconstruction. Utilizing scattered data points from the target object, we reconstruct a smooth, narrow volume by solving an Allen–Cahn-type equation enhanced with a control function. The non-negative attribute of this control function links the evolution of the main equation to an energy dissipation law, thereby ensuring energy stability. By developing and validating a leap-frog numerical scheme, we design a linear, second-order accurate, and unconditionally energy-stable method for advancing the solution in time. The spatial domain is discretized employing the finite difference method, with the fully discrete energy stability being analytically assessed. Through comprehensive numerical experiments, we confirm the algorithm’s accuracy, stability, and proficiency in reconstructing diverse 3D volumes. Furthermore, by evaluating the surfaces of reconstructed volumes, we identify how specific parameter selections influence the performance of the leap-frog numerical scheme.