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 → N be the vector of spectral coefficients and R → N the vector of N Galerkin constraints. A common two-part procedure is to first find all roots of R → N ( u → N ) = 0 → 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 → 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 → ( u → ; τ ) ≡ R → N + 1 ( u → ) − ( 1 − τ ) R → N + 1 ( u → N ) , τ ∈ [ 0 , 1 ] . Marching in small steps in τ , we proceed smoothly from the N -term to the N + 1 -term approximations.