{'fraction_non_alphanumeric': 0.01990632318501171, 'fraction_numerical': 0.0, 'mean_word_length': 5.526717557251908, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Achieving High-Order Convergence Rates with Deforming Basis Functions

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 computational 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.