Get Analytical and Numerical Aspects of Partial Differential PDF

By Etienne Emmrich, Petra Wittbold

ISBN-10: 3110204479

ISBN-13: 9783110204476

This article features a sequence of self-contained reports at the state-of-the-art in several components of partial differential equations, provided via French mathematicians. issues comprise qualitative houses of reaction-diffusion equations, multiscale tools coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation legislation.

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Additional resources for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series

Example text

Consider also the slope ω = dx of the discontinuity curve (more dt = u+ −u− exactly, the slope of its tangent line); notice that ω is equal to the value f ′ (u˜ ) at some point u˜ which lies strictly between u+ and u− . These three slopes satisfy the so-called Lax admissibility condition f ′ (u+ ) < ω = f (u+ ) − f (u− ) = f ′ (u˜ ) < f ′ (u− ). 7) Indeed, if f is strictly convex, then f ′ is a monotone increasing function, and the admissibility condition for this case of a convex flux function f ensures that u+ < u˜ < u− .

2) whatever be the flux function f = f (u)). Let us check the Rankine–Hugoniot condition on each of the three lines of discontinuity of the first kind (which are x = 0 and x = ±δt): as x = 0, we have u− = −δ , u+ = δ , and 2 dx δ 2 − (−δ ) f (u+ ) − f (u− ) =0= = ; dt δ − (−δ ) u+ − u− 28 Gregory A. Chechkin and Andrey Yu. Goritsky as x = −δt, we have u− = 0, u+ = −δ , and 2 f (u+ ) − f (u− ) dx (−δ ) − 02 = = −δ = ; dt u+ − u− (−δ ) − 0 as x = δt, we have u− = δ , u+ = 0, and 02 − δ 2 f (u+ ) − f (u− ) dx =δ= = .

7) again, since f ′ is a monotone decreasing function in this case. 7) is a particular case of the admissibility condition which is fundamental for the theory of systems of conservation laws. It was first formulated by the American mathematician P. D. Lax (see [30]). Therefore, we observe that, as t grows, the characteristics approach the discontinuity curve from both sides (see Fig. 10a); none of the two characteristics can move away from it (the case where the characteristics move away from the discontinuity curve as t grows is depicted in Fig.

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Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series by Etienne Emmrich, Petra Wittbold


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