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

Stability matrix for a linear system
Let us apply a Runge-Kutta scheme defined by A and w to the linear system
with y ∈ R m . Then one has
y1 = y0 + h
ν�
i=1
y ′ = By, y(0) = y0
wiBY (i) , Y (i) = y0 + h
ν�
aijBY (j)
Let e ≡ (1, . . . , 1) T ∈ R m denote a column vector whose components are unitary and let us
define the Kronecker products
⎛ ⎞
e ⊗ y n =
⎜
⎝
y n
y n
.
y n
⎟
⎠ , A ⊗ B =
⎛
⎜
⎝
j=1
a11B a12B · · · a1νB
a21B
.
a22B · · ·
...
a2νB
.
aν1B aν2B · · · aννB
After some manipulation the scheme can be conveniently written as
y n+1 = Ry n
where the m × m matrix of absolute stability R is given by
with Z ≡ h B.
R(Z) = Im + w T Z ⊗ (Iνm − A ⊗ Z) −1 e ⊗ Im,
⎞
⎟
⎠
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