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Mathematics of Computation

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


Author: Eitan Tadmor
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
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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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DOI: https://doi.org/10.1090/S0025-5718-1984-0758188-8
Article copyright: © Copyright 1984 American Mathematical Society

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