Front-pinned spectral collocation for the flame problem and related ignition problems

We consider global Chebyshev collocation for the ignition problems $y^{\mathrm{\prime }}=y^q(1-y)$, $y(0)=\delta$, $q\ge 2$; for $q=2$ this is the flame problem, a standard stiff test equation. After rescaling to $[-1,1]$ the solution has an interior layer of width $O(\mu )$, $\mu =(q-1){\delta }^{q-1}$, at a $\delta$-dependent position. Two obstacles arise: plain Chebyshev collocation needs $O(1\mathrm{/}\mu )$ nodes, and imposing the initial condition amplifies errors by a factor of order $1\mathrm{/}\mu$, which makes the collocation equations severely ill-conditioned. Our main contribution addresses the second one. Since all solutions are translates of one profile, the initial condition can be replaced by an equivalent interior condition. For an unknown $u=\phi (y)$, imposing it where the speed $\mathrm{\mid }{\phi }^{\mathrm{\prime }}{f}_q\mathrm{\mid }$ of $u$ along the orbit is maximal bounds the propagator of the linearized equation by one; for $u=\ln y$ this is $y(x^{\ast })=(q-1)\mathrm{/}q$. For the first obstacle we use the sinh map of Tee and Trefethen: the time function is explicit, the complex singularities of the profile lie exactly at $\pm (2k+1)\pi i$, and the map parameters follow in closed form, so that $O(\ln (1\mathrm{/}\mu ))$ nodes suffice. In double precision the pinned scheme reaches the interpolation error of the exact solution in five or six Newton iterations for $\mu$ down to ${10}^{-12}$, whereas the initial-condition form fails or converges to wrong discrete solutions, also in an integral (Birkhoff) reformulation. Computations in $256$-bit arithmetic confirm the predicted rate down to errors of ${10}^{-70}$, AAA approximation locates the singularities to within $2\mathrm{\%}$, and the method carries over to an Arrhenius reaction rate, with all parameters computed numerically.