Achieving High-Order Convergence Rates with Deforming Basis Functions

Abstract

This article studies the use of moving, deforming elliptical Gaussian basis functions to compute the evolution of passive scalar quantities in a two-dimensional, incompressible flow field. We compute an evolution equation for the velocity, rotation, extension and deformation of the com- putational elements as a function of flow quantities. We find that if one uses the physical flow velocity data calculated from the basis function centroid, the method has only second order spatial accuracy. However, by computing the residual of the numerical method, we can determine adjustments to the centroid data so that the scheme will achieve fourth-order spatial accuracy. Simulations with nontrivial flow parameters demonstrate that the methods exhibit the properties predicted by theory.