Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4967510 | Journal of Computational Physics | 2017 | 30 Pages |
Abstract
Walsh functions form an orthonormal basis set consisting of square waves. Square waves make the system well suited for detecting and representing functions with discontinuities. Given a uniform distribution of 2p cells on a one-dimensional element, it is proved that the inner product of the Walsh Root function for group p with every polynomial of degree â¤(pâ1) across the element is identically zero. It is also proved that the magnitude and location of a discontinuous jump, as represented by a Heaviside function, are explicitly identified by its Fast Walsh Transform (FWT) coefficients. These two proofs enable an algorithm that quickly provides a Weighted Least Squares fit to distributions across the element that include a discontinuity. It is shown that flux reconstruction relative to the FWT fit in partial differential equations provides improved accuracy. The detection of a discontinuity further enables analytic relations to locally describe its evolution and provide increased accuracy. Examples are provided for time-accurate advection, Burgers' equation, and quasi-one-dimensional nozzle flow.
Related Topics
Physical Sciences and Engineering
Computer Science
Computer Science Applications
Authors
Peter A. Gnoffo,