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
Frozen particle: a finite volume scheme...
We use a well-balanced finite volume scheme , preserving exactly
(some of) the stationary sols u(t, x) := u−1l {x0}.
Usual schemes are determined by a numerical flux g(·,·) :
– g locally Lipschitz;
– g(u, u) = u2
2 (consistency );
– g(·, b) is ր, g(a,·) is ց (monotonicity ).
We only modify g(·,·) at the interface x = 0 (between x0 and x1):
g −
λ
(a, b) = g(a, b + λ) and g+
λ (a, b) = g( a − λ,b).
Idea : g ±
λ “only see” the G 1 λ part of the germ !
Then the scheme writes
∀i = 0, 1 u n+1
i = u n i
i = 0 : u n+1
0 = un 0
i = 1 : u n+1
1
Numerical solution:
u∆(t, x) =
n∈N
− ∆t
∆x (g(un i , un i+1 )−g(un i−1 , un i ));
∆t − ∆x (g− λ (un 0 , un 1 )−g(un −1 , un 0 ));
= un 1 − ∆t
∆x (g(un 1, u n 2)−g +
λ (un 0, u n 1)).
i∈Z un i 1l (n∆t,(n+1)∆t)(t) 1l (xi−1/2,x i+1/2)(x).