Wavelet-based Multiresolution Particle Methods

Particle methods offer a robust numerical tool for solving transport problems across disciplines, such as fluid dynamics, quantitative biology or computer graphics. Their strength lies in their stability, as they do not discretize the convection operator, and appealing numerical properties, such as small dissipation and dispersion errors. Many problems of interest are inherently multiscale, and their efficient solution requires either multiscale modeling approaches or spatially adaptive numerical schemes. We present a hybrid particle method that employs a multiresolution analysis to identify and adapt to small scales in the solution. The method combines the versatility and efficiency of grid-based Wavelet collocation methods while retaining the numerical properties and stability of particle methods. The accuracy and efficiency of this method is then assessed for transport and interface capturing problems in two and three dimensions, illustrating the capabilities and limitations of our approach.