A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
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Mo<strong>de</strong>l <strong>and</strong> motivation Auxiliary steps Results h = 0: coupl<strong>in</strong>g h = 0: <strong>de</strong>f<strong>in</strong>ition, uniqueness h = 0: <strong>numerics</strong>, existence The coupled problem<br />
Convergence; existence of entropy solutions.<br />
Theorem (convergence of the scheme; existence of solutions)<br />
Assume u0 ∈ L ∞ (R). Then, un<strong>de</strong>r the CFL condition <strong>and</strong> assumption (H), the<br />
numerical scheme converges <strong>in</strong> L 1 loc(R + ×R) to the unique entropy solution to<br />
“<strong>Burgers</strong> with <strong>particle</strong>-at-zero” problem when ∆x tends to 0.<br />
In particular, the problem is well-posed , for L ∞ data <strong>and</strong> L 1 loc topology.<br />
Proof.<br />
• First assume that u0 ∈ BV(R).<br />
– BVloc bounds yield compactness: we get u an accumulation po<strong>in</strong>t of (u∆)∆ ;<br />
– well-balance property for (c−, c+) ∈ G 1 λ ∪ G 2 λ yields enough explicit<br />
stationary solutions v∆ to the scheme (at least, at the limit ∆x → 0);<br />
– us<strong>in</strong>g the approximate Kato <strong>in</strong>equalities on u∆ <strong>and</strong> the above special<br />
solutions v∆, at the limit we get Kato <strong>in</strong>equalities...<br />
but, these are precisely the adapted entropy <strong>in</strong>equalities !!<br />
– then u is (the unique) entropy solution (use caract. A. of entropy sols).<br />
• For the general case u0 ∈ L ∞ (R) , localize us<strong>in</strong>g f<strong>in</strong>ite speed of<br />
propagation; approximate u0 by BV(R)∩L 1 (R) functions(u n 0)n Use discrete<br />
L 1 contraction <strong>and</strong> the result of the BV case.