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, provided by way of French mathematicians. issues comprise 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 laws.
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Extra resources for Analytical and numerical aspects of partial differential equations : notes of a lecture series
12) with exactly two discontinuity lines. Indeed, such a solution would have two distinct jumps: a jump from the state 0 (on the left from the discontinuity line) to some constant state δ (on the right), and the jump from δ (now on 29 The Kruzhkov lectures the left) to 0 (now on the right). According to the Rankine–Hugoniot condition, these )−f (0) jumps can only occur along straight lines of the form x = f (δδ− t + C , C ∈ R. Since 0 the solution also obeys the zero initial datum, the constant C should be the same for the two jumps.
The latter means that the function obtained by the juxtaposition turns out to be continuous on the border ray x = ξt = u3 t, t > 0. 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.
The procedure of introducing the term εuxx into a first-order equation and the subsequent study of the limits of the solutions uε as ε → +0 is called the “vanishing viscosity” method. 9). Observe that we have ut = (εux − u2 /2)x ; thus we can introduce a potential U = U (t, x), determined from the equality dU = u dx + εux − u2 /2 dt. , the function U satisfies the equation 1 Ut + (Ux )2 = εUxx . 10), let us make the substitution U = −2ε ln z . Then Ut = −2ε zt , z Ux = −2ε zx , z Uxx = −2ε zxx (zx )2 + 2ε 2 .
Analytical and numerical aspects of partial differential equations : notes of a lecture series by Etienne Emmrich, Petra Wittbold