543 research outputs found
High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures
The work reported in this article presents a high-order, stable, and
efficient Gegenbauer pseudospectral method to solve numerically a wide variety
of mathematical models. The proposed numerical scheme exploits the stability
and the well-conditioning of the numerical integration operators to produce
well-conditioned systems of algebraic equations, which can be solved easily
using standard algebraic system solvers. The core of the work lies in the
derivation of novel and stable Gegenbauer quadratures based on the stable
barycentric representation of Lagrange interpolating polynomials and the
explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous
error and convergence analysis of the proposed quadratures is presented along
with a detailed set of pseudocodes for the established computational
algorithms. The proposed numerical scheme leads to a reduction in the
computational cost and time complexity required for computing the numerical
quadrature while sharing the same exponential order of accuracy achieved by
Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical
test examples to assess the efficiency and accuracy of the numerical scheme.
The present method provides a strong addition to the arsenal of numerical
pseudospectral methods, and can be extended to solve a wide range of problems
arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl
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