Multilevel Adaptive Particle Methods for Convection-Diffusion Equations

We present novel multilevel particle methods with extended adaptivity in areas where increased resolution is required. We present two complementary approaches as inspired by r-adaptivity and adaptive mesh refinement (AMR) concepts introduced in finite difference and finite element schemes. For the r-adaptivity a new class of particle based mapping functions is introduced while the particle-AMR method uses particle remeshing in overlapping domains as a key element. The advantages and drawbacks of the proposed particle methods are illustrated based on results from the solution of one-dimensional convection-diffusion equations while the extension of the method to higher dimensions is demonstrated in simulations of the inviscid evolution of an elliptical vortex.