A particle-in-Burgers model: theory and numerics - Laboratoire de ...

Model and motivation Auxiliary steps Results h = 0: coupling h = 0: definition, uniqueness h = 0: numerics, existence The coupled problem
Properties of the scheme...
Under the CFL condition: 2M∆t ∆x, (M being the Lipschitz constant of
the numerical flux g on the ad hoc interval of values of (u n i )n,i, the scheme
writes
∀i ∈ Z u n+1
i = Hi(u n i−1, u n i , u n i+1),
where functions Hi are monotoneր in each of the three arguments.
NB: since · ↦→ ·±λ are ր functions, monotonicity OK also for i = 0, 1.
Lemma (L ∞ bound — choice of M in the CFL condition)
Under the CFL condition, the scheme satisfies for all n ∈ N, i ∈ Z
min{ess inf
R − u0 −λ, ess inf
R + u0} u n i max{ess sup
Proposition (the scheme is (partially) well-balanced)
R −
u0, ess sup
R +
u0 +λ}.
(i) The initial datum v0(·) = c(·) = c−1l{x0},
(c−, c+) ∈ G 1 λ, is exactly preserved in the evolution by the scheme .
(ii) Let v∆ be the solution of the numerical scheme with
the initial datum v0(·) = c(·) = c−1l{x0},
(c−, c+) ∈ G 2 λ. Then v∆ converge to c in L 1 loc(R + ×R) as ∆x → 0.