A technique for accurate collocation residual calculations

A multiple-grid collocation method is presented that allows exact evaluation of residuals generated by truncated trial function expansion solutions to boundary-value problems with polynomial nonlinearities. The method is used to formulate a true, discrete analog to the Galerkin projection applicable to the same class of problems. The numerical techniques developed are used to study the convergence behavior of a nonlinear, reaction-diffusion problem as a function of Thiele modulus (phi) and trial function truncation number (N). The convergence problems encountered at high phi values are found to result from a second, physically meaningless solution to the modeling equations. This 'spurious' solution and the true solution are involved in a saddle-node bifurcation that limits the range of phi where solutions are found for most finite N; the solutions appear to asymptotically approach each other as phi, N --> infinity regardless of the discretization method. The saddle-stable manifold of the spurious solution also defines the boundary of the set of initial conditions that diverge during dynamic simulations prior to the saddle-node bifurcation; all initial conditions are found to diverge after this bifurcation point. (C) 1998 Elsevier Science S.A. All rights reserved.