Solutions of nonlinear differential equations with feature detection using fast Walsh transforms

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 ≤(p−1) 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.