A Crandall-Lions Error Estimate for a Fully Implicit Upwind Scheme with Lipschitz State Coupling
Abstract
We establish a Crandall-Lions-type error estimate for a fully implicit upwind approximation of a one-dimensional diagonal nonconservative hyperbolic system, without assuming strict hyperbolicity. The velocity field is Lipschitz on a compact rectangular state set, and the theorem is conditional on the existence of a globally Lipschitz componentwise viscosity solution and a discrete solution taking values in that set. The argument maximizes a single doubled functional over the component index and both signs. At its maximum, one signed component controls the full vector error, allowing the velocity difference to be estimated in the maximum norm. An exponential rescaling gives a short-time comparison, while a restarted barrier estimate handles contact at the initial time of each strip. Iterating the local estimate yields, on every finite time interval, an error of order O( √ ∆x + ∆t), up to the initial grid error. No sign condition on the state coupling, spatial monotonicity, or a global discrete gradient estimate is needed for the comparison itself.
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BibTeX
@misc{hajj2026crandalllions,
title = {{A Crandall-Lions Error Estimate for a Fully Implicit Upwind Scheme with Lipschitz State Coupling}},
author = {Ahmad El Hajj and Hassan Ibrahim},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05787502v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Crandall-Lions Error Estimate for a Fully Implicit Upwind Scheme with Lipschitz State Coupling
AU - Ahmad El Hajj
AU - Hassan Ibrahim
PY - 2026
DA - 2026/10/06
PB - HAL
UR - https://hal.science/hal-05787502v1
AB - We establish a Crandall-Lions-type error estimate for a fully implicit upwind approximation of a one-dimensional diagonal nonconservative hyperbolic system, without assuming strict hyperbolicity. The velocity field is Lipschitz on a compact rectangular state set, and the theorem is conditional on the existence of a globally Lipschitz componentwise viscosity solution and a discrete solution taking values in that set. The argument maximizes a single doubled functional over the component index and both signs. At its maximum, one signed component controls the full vector error, allowing the velocity difference to be estimated in the maximum norm. An exponential rescaling gives a short-time comparison, while a restarted barrier estimate handles contact at the initial time of each strip. Iterating the local estimate yields, on every finite time interval, an error of order O( √ ∆x + ∆t), up to the initial grid error. No sign condition on the state coupling, spatial monotonicity, or a global discrete gradient estimate is needed for the comparison itself.
ER -