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

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IMEX-RK methods An Implicit-Explicit (IMEX) Runge-Kutta scheme has the form Yi = �i−1 ν� 1 y0 + h ãijf(t0 + ˜cjh, Yj) + h aij ε g(t0 + cjh, Yj), y1 = y0 + h j=1 ν� i=1 ˜wif(t0 + ˜cih, Yi) + h j=1 ν� i=1 1 wi ε g(t0 + cih, Yi). Ã = (ãij), ãij = 0, j ≥ i and A = (aij): ν × ν matrices. Coefficient vectors: ˜c = (˜c1, . . . , ˜cν) T , ˜w = ( ˜w1, . . . , ˜wν) T , c = (c1, . . . , cν) T , w = (w1, . . . , wν) T . Double Butcher tableau: ˜c Ã ˜w T c A Sufficient condition to guarantee that f is always evaluated explicitly: the scheme for g is diagonally implicit (DIRK) and the first raw and first column of A are zero. Remarks • Similarly to splitting methods IMEX schemes can be applied as a sequence of single explicit steps for f and implicit steps for g. This property is important in applications. • Previously developed Runge-Kutta methods for similar problems can be cast in the IMEX formalism (Zhong methods, splitting methods). w T . 6
Some References: • F.Coron, B.Perthame, SIAM J. Numer. Anal., (1991) • R. Pember, SIAM J. Sci. Stat. Comp. and SIAM J. App. Math., (1993) • P. Roe, M. Arora, Num. Meth. PDEs, (1993). • S. Jin, J. Comp. Phys., (1995) • R. E. Caflisch, S. Jin and G. Russo, SIAM J. Numer. Anal., (1997). • U. Ascher, S. Ruuth, B. Wetton, SIAM J. Numer. Anal., (1995) • U. Asher, S. Ruuth, and R. J. Spiteri, Appl. Numer. Math., (1997) • X. Zhong, J. Comp. Phys., (1996) • G. Akridis, M. Crouzeix, C. Makridakis, Num. Math., (1999) • J. Frank, W. H. Hudsdorder, J. G. Verwer, App. Num. Math., (1997) • C. A. Kennedy, M. H. Carpenter, App. Num. Math., (2003) • L.P., G.Russo, Adv. Th. Comp. Math. (2000), preprint (2003) • M.L.Minion, Comm. Math. Sci. (to appear) • S.F.Liotta, V.Romano, G.Russo, SIAM J. Numer. Anal., (2001) • L.P., SIAM J. Numer. Anal, (2001). • M.K.Banda, A.Klar, L.P., M.Seaid, App. Math. Lett. (2003), preprint (2002) 7
Page 1 and 2: Joint work with Giovanni Russo HYP2
Page 3 and 4: Example: A simple prototype example
Page 5: We can consider the system of ode
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 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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