## The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme

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- Math. Comp.
**43**(1984), 353-368 Request permission

## Abstract:

We study the Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law ${u_t} + {f_x}(u) = 0$, where $f(u)$ is, say, strictly convex, $\ddot f \geqslant {\dot a_ \ast } > 0$. We show that the divided differences of the numerical solution at time*t*do not exceed $2{(t{\dot a_ \ast })^{ - 1}}$. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude, in sharp contrast to linear problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a

*quantitative*insight into the large-time behavior of the numerical computation.

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## Additional Information

- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp.
**43**(1984), 353-368 - MSC: Primary 65M05; Secondary 35L65
- DOI: https://doi.org/10.1090/S0025-5718-1984-0758188-8
- MathSciNet review: 758188