By Etienne Emmrich, Petra Wittbold
This article features a sequence of self-contained studies at the cutting-edge in several parts of partial differential equations, awarded via French mathematicians. issues comprise qualitative houses 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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Extra info for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series
We see that the jump between u− and u+ is admissible in the sense of the previous section if u− > u+ (respectively, u− < u+ ) and the graph of the flux function f is situated under the chord (respectively, above the chord) joining the points (u− , f (u−)) and (u+ , f (u+ )) (see Fig. 11). It turns out that the above reformulation of the admissibility rule for convex/concave flux functions remains appropriate for the case of an arbitrary flux function f . For a rather rigorous justification of this statement, let us use “physical” (more exactly, “fluid dynamics”) considerations based on the concepts of an ideal gas and a viscous gas.
To summarize, for the case of a convex or a concave flux function f = f (u), we have deduced the following condition for admissibility of discontinuities. Let u− , respectively u+ , be the one-sided limit of a generalized solution u = u(t, x) as the discontinuity curve is approached from the left, respectively from the right, along the x-axis. Then • • in the case of a convex function f = f (u) (for instance, f (u) = u2 /2, eu , . 1) may have jumps from u− to u+ only when u− > u+ ; in the case of a concave function f = f (u) (f (u) = −u2 , ln u, .
D. Lax (see ). 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. 10b). This means that those discontinuities are admissible which are due to the fact that characteristics of a smooth solution (smooth from each side of the discontinuity curve) tend to have intersections as t grows (the intersections eventually occur on the discontinuity curve).
Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series by Etienne Emmrich, Petra Wittbold