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

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---Pade 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 $$N_{feasible}$$Nfeasible. If the interesting branches can be resolved at least crudely by $$N_{feasible}$$Nfeasible degrees of freedom, then the solutions can be refined to high accuracy by DISH. However, solution branches that demand $$N > N_{feasible}$$N>Nfeasible 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.