Abstract
The nonlinear two-point boundary value problem (TPBVP for short) \[u_{xx}+u^{3}=0,\quad u(0)=u(1)=0,\] offers several insights into spectral methods.
First, it has been proved a priori that \[\int u(x)dx=\frac p{\sqrt{2}}.\] By building this constraint into the spectral approximation, the accuracy of \(N=1\) degrees of freedom is achieved from the work of solving a system with only N degrees of freedom. When N is small, generic polynomial system solvers, such as those in the computer algebra system Maple, can find all roots of the polynomial system, such as a spectral discretization of the TPBVP.
Our second point is that floating point arithmetic in lieu of exact arithmetic can double the largest practical value of N. (Rational numbers with a huge number of digits are avoided, and eliminating M symbols like \(\sqrt{2}\) and p reduces \(N+M\)-variate polynomials to polynomials in just the N unknowns.) Third, a disadvantage of an “all roots” approach is that the polynomial solver generates many roots \(( 3^N-1)\) -for our example – which are genuine solutions to the \(N\)-term discretization but spurious in the sense that they are not close to the spectral coefficients of a true solution to the TPBVP.
We show here that a good tool for “root-exclusion” is calculating \[\rho=\sqrt{\sum\limits_{n=1}^{N}b_{n}^{2}};\] spurious roots have \(\rho\) larger than that for the physical solution by at least an order of magnitude. The \(\rho\)-criterion is suggestive rather than infallible, but root exclusion is very hard, and the best approach is to apply multiple tools with complementary failings.
Authors
John P. Boyd
(Department of Climate & Space Sciences and Engineering, University of Michigan, United States)
Calin-Ioan Gheorghiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
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Cite this paper as
J.P. Boyd, C.I. Gheorghiu, All roots spectral methods: Constraints, floating point arithmetic and root exclusion, Applied Mathematics Letters 67 (2017) 28–32
DOI: 10.1016/j.aml.2016.11.015
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[1] J.P. Boyd, Tracing multiple solution branches for nonlinear ordinary differential equations: Chebyshev and Fourier spectral methods and a degree-increasing spectral homotopy [DISH], J. Sci. Comput. 19 (2016) 1113–1143.
[2] J.P. Boyd, Degree-increasing [N to N + 1] homotopy for Chebyshev and Fourier spectral methods, Appl. Math. Lett. 57 (2016) 77–81.
[3] C.I. Gheorghiu, D. Trif, The numerical approximation to positive solution for some reaction–diffusion problems, Pure Math. Appl.: Math. Optim. 11 (2001) 243–253.
[4] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, New York, 2001.
[5] C.-I. Gheorghiu, Spectral Methods for Differential Problems, Casa Cartii de Stiinta, Cluj-Napoca, Romania, 2007, 157 pp.
Available at http://www.ictp.acad.ro/gheorghiu/spectral.pdf.
[6] B.A. Finlayson, The Method of Weighted Residuals and Variational Principles, second ed., SIAM, New York, 2013.
[7] J.P. Boyd, Chebyshev and Legendre spectral methods in algebraic manipulation languages, J. Symbolic Comput. 16 (1993) 377–399.
[8] J.P. Boyd, Correcting three errors in Kantorovich & Krylov’s approximate methods of higher analysis: Energizing perturbation series and Chebyshev and legendre spectral algorithms with computer algebra, Amer. Math. Monthly 123 (2016) 241–257.