Front-pinned spectral collocation for the flame problem and related ignition problems
Abstract
We consider global Chebyshev collocation for the ignition problems y'=yq(1−y), y(0)=δ, q≥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(μ), μ=(q−1)δq−1, at a δ-dependent position. Two obstacles arise: plain Chebyshev collocation needs O(1/μ) nodes, and imposing the initial condition amplifies errors by a factor of order 1/μ, 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=φ(y), imposing it where the speed |φ'fq| of u along the orbit is maximal bounds the propagator of the linearized equation by one; for u=ln y this is y(x∗)=(q−1)/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 ±(2k+1)πi, and the map parameters follow in closed form, so that O(ln(1/μ)) nodes suffice. In double precision the pinned scheme reaches the interpolation error of the exact solution in five or six Newton iterations for μ 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%, and the method carries over to an Arrhenius reaction rate, with all parameters computed numerically.
Related
- Advancements in Spectral Collocation Methods for High-Order Eigenvalue Problems
- CQ-collocation BEMs for exterior wave propagation problems: the 3D Neumann case
- Well-Conditioned Birkhoff-Collocation Methods for Elliptic-type Problems in Multiple Dimensions
- A Shifted Balanced Polynomial Collocation Framework for Variable-Coefficient Multi-Term Time-Fractional Convection–Diffusion–Reaction Problems
- Two-Parameter General-Shift Lucas Polynomial Spectral Collocation for Time-Fractional Nonlinear Partial Integro-Differential Equations
BibTeX
@misc{boros2026frontpinned,
title = {{Front-pinned spectral collocation for the flame problem and related ignition problems}},
author = {Imre Boros},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27864},
eprint = {2610.27864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27864},
}
RIS
TY - UNPB
TI - Front-pinned spectral collocation for the flame problem and related ignition problems
AU - Imre Boros
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27864
UR - https://arxiv.org/abs/2610.27864
AB - We consider global Chebyshev collocation for the ignition problems y'=y^q(1−y), y(0)=δ, q≥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(μ), μ=(q−1)δ^q−1, at a δ-dependent position. Two obstacles arise: plain Chebyshev collocation needs O(1/μ) nodes, and imposing the initial condition amplifies errors by a factor of order 1/μ, 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=φ(y), imposing it where the speed |φ'f_q| of u along the orbit is maximal bounds the propagator of the linearized equation by one; for u=ln y this is y(x^∗)=(q−1)/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 ±(2k+1)πi, and the map parameters follow in closed form, so that O(ln(1/μ)) nodes suffice. In double precision the pinned scheme reaches the interpolation error of the exact solution in five or six Newton iterations for μ 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%, and the method carries over to an Arrhenius reaction rate, with all parameters computed numerically.
ER -