Tamtam Proceedings - lamsin

A method for optimal control problems 77control problem with a discontinuous value function in dimension 2.2. The UltraBee schemeFirst we consider a linear advection equation with positive constant velocity a.v t (x, t) + a.v x (x, t) = 0, v(x, 0) = v 0 (x), x ∈ IR, t ∈ IR + .Let us introduce some usefull notations for the discretization of the transport problem.∆t will express the time step, ∆x the space step of a regular grid G and ν j the local CFLnumber at cell jν j = a∆t∆x .The UltraBee scheme is a finite volume scheme of typev n+1j= vj n − ν j (v n,L − v n,R ), vj+ 1 2 j− 1 j 0 =2∫ xj+ 12x j− 12v 0 (x)dx, ∀j ∈ IZ, (2)∆xwhere v n,L and v n,R are numerical flux between cells j and j + 1. When the velocity isj+ 1 2 j+ 1 2positive and does not change sign v n,Lj+ 1 2= v n,R = v n , wherej+ 1 j+ 1 22v n j+ 1 2= v n,L = v n,R = vj+ 1 2 j+ 1 j n + 12 2 (1 − ν j)ρ n j+(v 1 j+1 n − vj n ); (3)2with ρ n j+ 1 2= ρ(r n , νj+ 1 j ) = max[0, min( 2 ν2jr n 2,j+ 1 1−ν2 j)], and r n j+ 1 2= vn j −vn j−1v.j+1 n −vn jLet us introduce some additional notations:{ mn= min(vj+ 1 j n, vn j+1 ) , M n = max(v2j+ 1 j n, vn j+1 ),2b n,+j = 1 ν j(v n j − M n j− 1 2) + M n j− 1 2, B n,+j = 1 ν j(vj n − mn ) + m n .j− 1 2 j− 1 2Désprès and Lagoutière proved that the v n j+ 1 2defined in [3] is also given bymin |v n j+ 1 2− vj+1| n under constraint b n,+j ≤ v n j+≤ B n,+1 j .2Hence, the constraint being linear, the flux takes just three values (always in the casea > 0)⎧= b ⎪⎨n,+j if vj+1 n ≤ bn,+ j ,⎪⎩v n j+ 1 2v n j+ 1 2v n j+ 1 2= v n j+1 if b n,+j= B n,+j if B n,+≤ vj+1 n ≤ Bn,+ j ,(4)j ≤ vj+1 n .TAMTAM –Tunis– 2005