Optimizing performance of the deconvolution model reduction for large ODE systems

Abstract

We investigate the numerical performance of the regularized deconvolution closure introduced recently by the authors. The purpose of the closure is to furnish constitutive equations for Irving-Kirkwood-Noll procedure, a well known method for deriving continuum balance equations from the Newton’s equations of particle dynamics. A version of this procedure used in the paper relies on spatial averaging developed by Hardy, and independently by Murdoch and Bedeaux. The constitutive equations for the stress are given as a sum of nonlinear operator terms acting on the mesoscale average density and velocity. Each term is a “convolution sandwich” containing the deconvolution operator, a composition or a product operator, and the convolution (averaging) operator. Deconvolution is constructed using filtered regularization methods from the theory of ill-posed problems. The purpose of regularization is to ensure numerical stability. The particular technique used for numerical experiments is truncated singular value decomposition (SVD). The accuracy of the constitutive equations depends on several parameters: the choice of the averaging window function, the value of the mesoscale resolution parameter, scale separation, the level of truncation of singular values, and the level of spectral filtering of the averages. We conduct numerical experiments to determine the effect of each parameter on the accuracy and efficiency of the method. Partial error estimates are also obtained.