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

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).
à = (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
.
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