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Tracing Multiple Solution Branches for Nonlinear Ordinary Differential Equations: Chebyshev and Fourier Spectral Methods and a Degree-Increasing Spectral Homotopy [DISH]

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Abstract

Nonlinear differential and integral equations often have multiple solution branches. Reliable sofware to find all roots of a system of polynomial equations, as generated by standard discretizations for differential equations with polynomial nonlinearity, are now widely available. It is now feasible to pursue a global, all-branches attack on differential and integral equations. Unfortunately, both the number of solutions and computational cost grow exponentially fast with N, the number of degrees of freedom. And what if the nonlinearity is not polynomial? And how to continue from small-N to large N? Here, we show that Chebyshev and Fourier Petrov–Galerkin methods are “N-minimizing”, but especially with exploitation of cryptoperiodicity, parity and other symmetries, basis recombination, and unconventional Gegenbauer polynomial Petrov–Galerkin weights. We also show that Chebyshev–Padé technology is efficient at “polynomializing” transcendental nonlinearity. If the small-N solution is sufficiently accurate, continuation to larger N is a simple matter of initializing Newton’s iteration for large N with the solution for small N (“modal persistence”). When this fails, we introduce Newton-DISH: a Degree-Increasing Spectral Homotopy which is slower but more resilient. The major limitation is that present-day all-branches polynomial system solvers are limited, due to both operation count and memory storage, to rather modest N, typically 4–20, which we dub \(N_{feasible}\). If the interesting branches can be resolved at least crudely by \(N_{feasible}\) degrees of freedom, then the solutions can be refined to high accuracy by DISH. However, solution branches that demand \(N > N_{feasible}\) for even a poor approximation lie outside the scope of the strategy described here. Our theme is not that the “all-roots” strategy is universally applicable, but rather that its domain of usefulness is greatly expanded by using spectral methods in lieu of low order finite differences and finite elements. We demonstrate the power of the polynomial-solver and Chebyshev partnership by finding a hitherto unsuspected second branch (smooth and real-valued) of the Sag-Haselgrove nonlinear integral equation, which has been studied for more than half a century.

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Notes

  1. Allgower et al. add-a-point homotopy is symmetry-destroying unless the numerical domain is changed to half the physical domain with a homogeneous Neumann boundary condition imposed at the symmetry point.

  2. Although the authors do not comment, this differential equation describes the time evolution of a pendulum and can be solved exactly in terms of elliptic functions.

  3. The appellation “Eagle’s constant” arose several decades ago in response to Eagle’s book Elliptic Functions As They Should Be [21] where he advocated introducing a new, never-to-be-used-for-anything-else symbol for \( \pi /2\). Eagle independently invented B-splines (which he called “lath functions” [20]) sixteen years before Schoenberg, who is usually given the credit.

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Acknowledgments

This work was supported by the National Science Foundation through Grants OCE 1059703 and DMS 1521158. I thank Charles Wampler and Jonathan Hauenstein for helpful conversations and email. I thank the reviewer for useful comments; the applicability of the method only to branches resolvable by small N is more explicit in response.

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  1. Department of Climate and Space Sciences and Engineering, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI, 48109, USA

    John P. Boyd

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Boyd, J.P. Tracing Multiple Solution Branches for Nonlinear Ordinary Differential Equations: Chebyshev and Fourier Spectral Methods and a Degree-Increasing Spectral Homotopy [DISH]. J Sci Comput 69, 1115–1143 (2016). https://doi.org/10.1007/s10915-016-0229-2

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