Anisotropic Kernels for Particle Flow Simulation

Abstract

This contribution discusses the construction and the utility of anisotropic kernels for numerical fluid flow simulation. So far, commonly used radial kernels, such as Gaussians, (inverse) multiquadrics and polyharmonic splines, were proven to be powerful tools in various applications of multivariate scattered data approximation. Due to the well-known uncertainty principle, however, their resulting reconstruction methods are often critical when it comes to combine high order approximation with numerical stability. In many cases this leads to severe limitations, especially when it comes to fluid flow simulations. Therefore, more sophisticated kernel methods are required. In this paper, we show how to obtain anisotropic positive definite kernels from standard kernels rather directly. Our proposed construction yields a new class of more flexible kernels that are particularly useful for fluid flow simulations. To this end, the finite volume particle method is used as a prototype of our discussion, where scattered data approximation is needed in the recovery step of weighted essentially non-oscillatory (WENO) reconstructions. Supporting numerical examples and comparisons are provided.