A discrete Aronson–Bénilan estimate for an upwind growth scheme
Fix $X, T, p_H > 0$ and $G \in C^1([0, p_H])$ with $G(p_H) = 0$ and $G'(p) \le -\alpha < 0$. On the grid $x_i = ih$ in $[-X, X]$, with $h = X/M$ and $-M \le i \le M$, consider nonnegative densities $n_i(t)$, pressures $p_i = n_i^\gamma$ with $\gamma > 1$, and the semi-discrete upwind scheme of David and Ruan for the tumour growth model $\partial_t n - \partial_x(n\, \partial_x p) = n G(p)$, $$\dot n_i = \frac{n_{i+1/2}\, q_{i+1/2} - n_{i-1/2}\, q_{i-1/2}}{h} + n_i G(p_i), \qquad q_{i+1/2} = \frac{p_{i+1} - p_i}{h},$$ $$n_{i+1/2} = \begin{cases} n_i, & q_{i+1/2} \le 0, \\ n_{i+1}, & q_{i+1/2} > 0, \end{cases}$$ with the reflected Neumann ghost values $n_{-M-1} = n_{-M+1}$ and $n_{M+1} = n_{M-1}$, and the same pressure law at the ghost nodes. Assume $0 \le p_i(0) \le p_H$ and that, for a fixed $C_0$ independent of $h$ and $\gamma$, each of $$h \sum_i n_i(0), \qquad h \sum_i p_i(0), \qquad \sum_{i=-M}^{M-1} |n_{i+1}(0) - n_i(0)|, \qquad h \sum_i |\dot n_i(0)|$$ is at most $C_0$, where unrestricted sums run over $-M \le i \le M$ and $\dot n_i(0)$ is given by the scheme. Write $$w_i(t) = \frac{p_{i+1}(t) - 2p_i(t) + p_{i-1}(t)}{h^2} + G(p_i(t)).$$ For $G(p) = \alpha(p_H - p)$ with $\alpha \ge 0$, David and Ruan proved $w_i \ge -1/(\gamma t)$ for this scheme when $\gamma = 1$, and derived it by a formal argument in the limit $\gamma \to \infty$.
Problem. Is there a constant $C = C(X, T, p_H, G, C_0)$, independent of $h$ and of $\gamma > 1$, such that every such solution satisfies $$w_i(t) \ge -\frac{C}{\gamma t} \qquad (-M \le i \le M,\ 0 < t \le T)?$$ The question concerns this fixed-grid upwind scheme, not another discretisation of the same equation.
Source: David and Ruan (2022), Section 2.3, who leave open the discrete Aronson–Bénilan estimate for γ > 1 and a general growth term G. Stated by AIM — Open Applied Problems as a bound uniform in the mesh and in γ.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Aronson–Bénilan estimate, porous medium equation, upwind schemes
References
@article{david2022,
author = {David, Noemi and Ruan, Xinran},
title = {An asymptotic preserving scheme for a tumor growth model of porous medium type},
journal = {ESAIM Math. Model. Numer. Anal.},
volume = {56},
number = {1},
year = {2022},
pages = {121--150},
doi = {10.1051/m2an/2021080},
eprint = {2105.10376},
archivePrefix = {arXiv},
}
@misc{aim-upwind-discrete-aronson-benilan,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A discrete Aronson–Bénilan estimate for an upwind growth scheme}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/505-upwind-discrete-aronson-benilan.md},
note = {Entry 505 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
The uniform estimate fails with fixed data bounds and G(p)=1-p. The proof selects gamma after the mesh; the numerical experiments are corroboration, not a uniform choice of gamma.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, siavash-sadeghi-505/aim505_report.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00734, "A discrete Aronson–Bénilan estimate for an upwind growth scheme", clopen.solutions.