Analytical and Numerical Aspects of Partial Differential by Etienne Emmrich, Petra Wittbold

Analytical and Numerical Aspects of Partial Differential by Etienne Emmrich, Petra Wittbold

By Etienne Emmrich, Petra Wittbold

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

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40) by ′ E (u). 41) since E ′′ (u) 0 and ε > 0. 11 and integrate it over ΠT . Using the integrationby-parts formula, we transfer all the derivatives to the test function ϕ: u − ΠT f ′ (ξ )E ′ (ξ ) dξ dx dt ϕt E (u) + ϕx k ε ΠT ϕxx E (u) dx dt 49 The Kruzhkov lectures Passing to the limit as ε → +0, we get u ΠT f ′ (ξ )E ′ (ξ ) dξ dx dt ϕt E (u) + ϕx 0. 42) k Let {Em } be a sequence of C 2 -functions approximating the function u → |u − k| uniformly on R. 42) and pass to the limit ′ ′ as m → ∞. We can choose Em in such a way that Em is bounded and Em (ξ ) → sign(ξ − k ) for all ξ ∈ R, ξ = k .

7); this equality of the two areas is a direct consequence of the Rankine–Hugoniot condition. Figure 7. Area-preserving “overturning” of the graph. 7. 2). Denote +∞ S (t) = u(t, x) dx. , S (t) ≡ const. Proof. Indeed, we can write x(t) S (t) = +∞ u(t, x) dx + u(t, x) dx, x(t) −∞ where x = x(t) is the curve of discontinuity of the generalized solution u = u(t, x). As previously, we denote by u± = limx→x(t)±0 u(t, x) the one-sided limits (limits along the x-axis) of the solution u on the discontinuity curve.

Consequently, here the discontinuity is a weak, not a strong one. Now we can solve completely the Riemann problem for the Hopf equation. Here, two substantially different situations should be considered: (i) When u− > u+ , we can construct a shock wave solution, where the two constants u− and u+ are joined across the ray x = u2 +2 u1 t, according to the Rankine– Hugoniot condition (see Fig. 16): u(t, x) = u− for x < u+ for x > u− +u+ t, 2 u− +u+ t. 4) 52 Gregory A. Chechkin and Andrey Yu. Goritsky Figure 16.

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