Global field interpolation for particle methods

When a particle representation is used as a numerical discretization of a continuum problem, one often represents the field variable as a linear combination of moving basis functions (the particles). In this Lagrangian approach, the particles can become severely strained, which for continuum representations results in a deterioration of spatial accuracy. Thus, most workers face the need to remediate sets of particles so that the basis functions do not cease to overlap in space. While various investigators have developed means for replacing a set of particles with another that is more uniform, this subject is rarely addressed in the literature. The more general problem is that of finding a set of basis functions such that the field of interest is well represented by their superposition. We refer to this procedure as field interpolation. One approach is to formulate this as a radial basis function interpolation problem, which with global methods, will result in a dense and ill-conditioned system that needs to be solved. In particle simulation of fluid flow, a recent advance incorporates deforming basis functions to provide fourth-order accuracy in space. The requirement of an accurate field interpolation scheme becomes more stringent in this context, to maintain the overall high order. We have developed methods for performing field interpolation in this application, using custom preconditioning with iterative methods on the one hand, and deblurring techniques from image processing on the other.