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

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Theorem guarantees that in the stiff limit the numerical scheme becomes the explicit RK scheme applied to the equilibrium system, and therefore the order of accuracy of the limiting scheme is greater or equal to the order of accuracy of the original IMEX scheme. In particular this implies that if the explicit part of the IMEX scheme is SSP then, in the stiff limit, we will obtain an SSP method for the limiting conservation law. This asymptotic SSP property is essential to avoid spurious oscillations in the limit scheme for the limiting system of conservation laws. We recall that if U n represents a vector of solution values (for example obtained from a method of lines approach) we recall the following Definition 2 A sequence {U n }n∈N is said to be strongly stable in a given norm ||·|| provided that ||U n+1 || ≤ ||U n || for all n ≥ 0. The most commonly used norms are the T V -norm and the infinity norm. 16
Examples of IMEX-SSP schemes In all these schemes the implicit tableau corresponds to an L−stable scheme, that is w T A −1 e = 1, e being a vector whose components are all equal to 1, whereas the explicit tableau is SSPk, where k denotes the order of the SSP scheme. IMEX-SSP2(2,2,2) L-stable scheme 0 0 0 1 1 0 1/2 1/2 IMEX-SSP2(3,2,2) stiffly accurate scheme 0 0 0 0 0 0 0 0 1 0 1 0 0 1/2 1/2 IMEX-SSP2(3,3,2) stiffly accurate scheme 0 0 0 0 1/2 1/2 0 0 1 1/2 1/2 0 1/3 1/3 1/3 γ γ 0 1 − γ 1 − 2γ γ 1/2 1/2 1/2 1/2 0 0 0 −1/2 1/2 0 1 0 1/2 1/2 0 1/2 1/2 γ = 1 − 1 √ 2 1/4 1/4 0 0 1/4 0 1/4 0 1 1/3 1/3 1/3 1/3 1/3 1/3 17
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: Remark • Clearly one may claim th
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 and 34: Accuracy Test, Convergence Rates ε
Page 35 and 36: Monatomic gas in Extended Thermodyn
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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