Signal processing approach to mesh refinement in simulations of axisymmetric droplet dynamics

We propose a novel mesh refinement scheme based on signal processing for boundary integral simulations of inviscid droplet dynamics with axial symmetry. A key idea is to directly access the Fourier coefficients of a principal curvature as a function of the arclength through a natural change of variables. The trapezoidal rule is applied to those Fourier-type integrals and the resulting formula fits in the framework of the non-uniform fast Fourier transform. This observation enables to efficiently use an envelope analysis and smoothing filter to generate guidelines for mesh refinement in two singularity formation scenarios. Applications also include a non-iterative construction of the uniform parametrization for an important class of plane curves, which is used in a convergence study of the time-stepping procedure implemented in the previous work by Nitsche and Steen [J. Comput. Phys. 200 (2004) 299].