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 />
Properties of the scheme...<br />
Un<strong>de</strong>r the CFL condition: 2M∆t ∆x, (M be<strong>in</strong>g the Lipschitz constant of<br />
the numerical flux g on the ad hoc <strong>in</strong>terval of values of (u n i )n,i, the scheme<br />
writes<br />
∀i ∈ Z u n+1<br />
i = Hi(u n i−1, u n i , u n i+1),<br />
where functions Hi are monotoneր <strong>in</strong> each of the three arguments.<br />
NB: s<strong>in</strong>ce · ↦→ ·±λ are ր functions, monotonicity OK also for i = 0, 1.<br />
Lemma (L ∞ bound — choice of M <strong>in</strong> the CFL condition)<br />
Un<strong>de</strong>r the CFL condition, the scheme satisfies for all n ∈ N, i ∈ Z<br />
m<strong>in</strong>{ess <strong>in</strong>f<br />
R − u0 −λ, ess <strong>in</strong>f<br />
R + u0} u n i max{ess sup<br />
Proposition (the scheme is (partially) well-balanced)<br />
R −<br />
u0, ess sup<br />
R +<br />
u0 +λ}.<br />
(i) The <strong>in</strong>itial datum v0(·) = c(·) = c−1l{x0},<br />
(c−, c+) ∈ G 1 λ, is exactly preserved <strong>in</strong> the evolution by the scheme .<br />
(ii) Let v∆ be the solution of the numerical scheme with<br />
the <strong>in</strong>itial datum v0(·) = c(·) = c−1l{x0},<br />
(c−, c+) ∈ G 2 λ. Then v∆ converge to c <strong>in</strong> L 1 loc(R + ×R) as ∆x → 0.