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Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
A simple initial choice for basis functions is the so-called monomial basis. Essentially, in 1-D, these basis functions to order N are φn(r)=rn,wheren=0,…N
Parametric PGD model used with orthogonal polynomials to assess efficiently the building's envelope thermal performance
Published in Journal of Building Performance Simulation, 2021
Marie-Hélène Azam, Julien Berger, Sihem Guernouti, Philippe Poullain, Marjorie Musy
According to Trefethen (2013), the monomial basis is comfortable but should never be used to approximate a function. If we compare the condition number for inversion of the three basis, the Chebyshev and Legendre polynomials basis have a smaller condition number than the monomial one. If the condition number of a matrix is large, the matrix is close to being singular. The condition number reveals that the projection of the field of interest on the monomial basis will be sensitive to numerical round-off errors and perturbations in the input data. Moreover, monomial basis do not meet sparsity condition as its coefficients increase with the order. Therefore, this basis should not be used here to parameterize the initial condition.
Accurate Compression of Tabulated Chemistry Models with Partition of Unity Networks
Published in Combustion Science and Technology, 2022
Elizabeth Armstrong, Michael A. Hansen, Robert C. Knaus, Nathaniel A. Trask, John C. Hewson, James C. Sutherland
To each partition, a set of basis functions, , with basis coefficients, denoted , are then assigned to provide a localized reconstruction of the data. In this work, we employ as bases the space of tensor product of univariate Taylor monomials, which creates a polynomial of user-specified maximum degree. For example, in two-dimensions, a quadratic basis consists of monomial basis functions . Our demonstrations do not exceed basis degree two.