Implicit-Explicit Runge-Kutta schemes for hyperbolic systems ... - utenti

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Initial layer fix Next, we test the schemes with following two Riemann problems ρ(x,t) 2.05 2 ρl = 2, ml = 1, zl = 1, x < 0.2, ρr = 1, mr = 0.13962, zr = 1, x > 0.2, ρl = 1, ml = 0, zl = 1, x < 0, ρr = 0.2, mr = 0, zr = 1, x > 0. ε=1e−008, t=0.50, N=200 1.95 −0.2 −0.15 −0.1 −0.05 0 x 0.05 0.1 0.15 0.2 −1 −2 −3 −4 x ε=1e−008, 10−3 5 4 3 2 1 0 t=0.25, N=200 −5 0.2 0.25 0.3 0.35 x 0.4 0.45 0.5 Initial layer for ρ in problem 1 (left) and departure from equilibrium z − zE in problem 2 (right) for ε = 10 −8 . Left problem: ARS(2,2,2) (∗), ARSF(2,2,2) (◦), IMEX-SSP2(2,2,2) (×). Right problem: IMEX- SSP2(2,2,2) (+), IMEX-SSP2F(2,2,2) (×), ARS(2,2,2) (◦). z(x,t)−z E (x,t) 29 (1) (2)
Monatomic gas in Extended Thermodynamics U = ⎛ ⎜ ⎝ ρ ρv 1 2ρv2 + 3 Ut + F (U)x = 1 ɛ R(U) 2 p 2 3 ρv2 + σ ρv 3 + 5vp + 2σv + 2q F = ⎛ ⎜ ⎝ ⎞ ⎛ ⎟ ⎜ ⎟ ⎜ ⎟ , R = − ⎜ ⎠ ⎝ ρv ρv2 + p + σ vp + σv + q 2 3 0 0 0 ρσ ρ(2q + 3vσ) 1 2ρv3 + 5 2 2 3ρv3 + 4 7 8 vp + vσ + 3 3 15q ρv4 + 5 p2 32 + 7σp + ρ ρ 5 qv + v2 (8p + 5σ) ρ: density, u: velocity, p: pressure, σ: stress, q: heat flux As ɛ → 0 ⇒ σ → 0, q → 0 we obtain the Euler equations for monatomic gas. We use IMEX-SSP2(2,2,2) central scheme for a generalization of classical Sod’s problem U = Ul = (1, 0, 5, 0, 0), x < 0.5, U = Ur = (0.125, 0, 0.5, 0, 0), x > 0.5. ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ , 30
Page 1 and 2: Joint work with Giovanni Russo HYP2
Page 3 and 4: Example: A simple prototype example
Page 5 and 6: We can consider the system of ode
Page 7 and 8: Some References: • F.Coron, B.Per
Page 9 and 10: Higher order: IMEX schemes can be c
Page 11 and 12: CJR(3,2,2) β = LRR(3,2,2) ARS(2,2,
Page 13 and 14: Asymptotic properties of IMEX schem
Page 15 and 16: Remark • Clearly one may claim th
Page 17 and 18: Examples of IMEX-SSP schemes In all
Page 19 and 20: Stability analysis When studying th
Page 21 and 22: The corresponding scalar function R
Page 23 and 24: ξ h 5 4 3 2 1 0 −1 −2 −3 −
Page 25 and 26: ξ h 5 4 3 2 1 0 −1 −2 −3 −
Page 27 and 28: Numerical test Prototype of stiff s
Page 29 and 30: Convergence rates of some second an
Page 31 and 32: IMEX schemes cannot be applied stra
Page 33: Accuracy Test, Convergence Rates ε
Page 37 and 38: Lattice-Boltzmann models ∂tf + v
Page 39 and 40: Navier-Stokes case (τ = 0.01): Fir

Implicit-Explicit Runge-Kutta schemes for hyperbolic systems ... - utenti

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