Implicit-Explicit Runge-Kutta schemes for hyperbolic systems ... - utenti
Implicit-Explicit Runge-Kutta schemes for hyperbolic systems ... - utenti
Implicit-Explicit Runge-Kutta schemes for hyperbolic systems ... - utenti
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Lattice-Boltzmann models<br />
∂tf + v<br />
ɛ · ∇xf = 1<br />
ɛ 2 τ (f − f eq ), x, v ∈ R 2 ,<br />
v ∈ {c0, . . . , cN−1}, N = 9, ci ∈ {(0, 0), (0, ±1), (±1, 0), (±1, ±1)}.<br />
f eq �<br />
[ρ, u](v) = ρ 1 + 3u · v − 3<br />
2 |u|2 + 9<br />
�<br />
(u · v)2 f<br />
2 ∗ (v),<br />
ρ(x, t) =<br />
N−1 �<br />
i=0<br />
f(x, ci, t), ρu(x, t) =<br />
N−1 �<br />
i=0<br />
cif(x, ci, t),<br />
f ∗ (c0) = 4<br />
9 , f ∗ (ci) = 1<br />
9 , i = 1, ..., 4, f ∗ (ci) = 1<br />
, i = 5, ..., 8<br />
36<br />
As ɛ → 0 we obtain the incompressible Navier-Stokes equations with Reynolds number<br />
O(1/τ).<br />
shear layer<br />
(x, y) ∈ [0, 2π] 2 , ux(x, 0) = 0.05 sin(x), periodic b.c. and<br />
uy(x, y) = tanh(15(y − π/2)/π), y < π, uy(x, y) = tanh(15(3π/2 − y)/π), y > π.<br />
We test IMEX-SSP2(2,2,2) and IMEX-SSP3(3,3,2) <strong>schemes</strong> combined with second and<br />
third order upwind <strong>schemes</strong> based on CWENO reconstructions.<br />
32