45th NIA CFD Seminar

Topic: Walsh Functions in Numerical Simulation: A New Framework for Solving Nonlinear Systems of PDEs

Date: Tuesday, April 22, 2014

Time: 11:00am-noon (EST)

Room: NIA, Rm141

Speaker: Peter Gnoffo

Speaker Bio: Dr. Peter Gnoffo is Senior Computational Aerothermodynamicist at NASA Langley Research Center. He earned his Ph.D in Mechanical and Aerospace Engineering at Princeton University in 1983. He joined NASA in 1974 after receiving a B.S. in Aerospace Engineering from Polytechnic Institute of Brooklyn. The subject lecture follows recent work published in the Journal of Computational Physics, Vol 258, pp 650-688, Feb 2014, titled ‘Global Series Solutions of Nonlinear Differential Equations with Shocks Using Walsh Functions.’

Abstract: A segmented, orthonormal basis function set composed of Walsh Functions is used for deriving global solutions (valid over the entire domain) to nonlinear differential equations that include discontinuities. A powerful, self-mapping characteristic of this set is closure under multiplication — the product of any two elements of the set is also an element of the set: gn(x) gm(x) = gk(x) (xb – xa)^{-1/2}. In the same way that Fourier series are used to generate global solutions to linear problems, this self-mapping property under multiplication allows similar approaches to non-linear problems. A new derivation of the basis functions applies a fractal-like algorithm (infinitely self-similar) focused on the distribution of segment lengths. Only two segment lengths are allowed in a group p. A recursive-folding algorithm that propagates fundamental symmetries to successive functions in the series determines the distribution of segment lengths. Functions, including those with discontinuities, may be represented on the domain as a series in gn(x) with no occurrence of a Gibbs phenomenon (ringing) across the discontinuity. Integrals and derivatives are computed using simple matrix multiplication of series representations of functions without the need for divided differences. A FORTRAN module for supporting Walsh function simulations is discussed. Examples are discussed for solution of the time dependent problems: an advection equation, a Burgers equation, and a Riemann problem.

Additional information, including the webcast link, can be found at the NIA CFD Seminar website, which is temporarily located at

http://www.hiroakinishikawa.com/niacfds/index.html