A degree-increasing [N to N+1] homotopy for Chebyshev and Fourier spectral methods

Hitherto, as a tool for tracing all branches of nonlinear differential equations, resolution-increasing homotopy methods have been applied only to finite difference discretizations. However, spectral Galerkin algorithms typically match the error of fourth order differences with one-half to one-fifth the number of degrees of freedom N in one dimension, and a factor of eight to a hundred and twenty-five in three dimensions. Let (u) over right arrow (N) be the vector of spectral coefficients and (R) over right arrow (N) the vector of N Galerkin constraints. A common two-part procedure is to first find all roots of (R) over right arrow (N)((u) over right arrow (N)) = (0) over right arrow using resultants, Groebner basis methods or block matrix companion matrices. (These methods are slow and ill-conditioned, practical only for small N.) The second part is to then apply resolution-increasing continuation. Because the number of solutions is an exponential function of N, spectral methods are exponentially superior to finite differences in this context. Unfortunately, (u) over right arrow (N) is all too often outside the domain of convergence of Newton's iteration when N is increased to (N+1). We show that a good option is the artificial parameter homotopy (H) over right arrow((u) over right arrow;tau) = (R) over right arrow (N+1) ((u) over right arrow) - (1 -r)(R) over right arrow (N+1)((u) over right arrow (N)), tau is an element of [0, 1]. Marching in small steps in tau, we proceed smoothly from the N-term to the N + 1-term approximations. (C) 2016 Elsevier Ltd. All rights reserved.