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

The corresponding scalar function
R(z), z ∈ R
is said the function of absolute stability. The eigenvalues of the matrix of absolute
stability are given by the absolute stability function evaluated at the eigenvalues of the
matrix Z
λ(R(Z)) = R(λ(Z))
and therefore the spectral radius of the matrix of absolute stability is given by
ρ(R(Z)) = m
max
j=1 |R(λj(Z))|.
Similarly for a partitioned Runge-Kutta scheme applied to the system
one obtains
y ′ = B1y + B2y
y n+1 = R(Z1, Z2)y n
where Z1 = h B1, Z2 = h B2, and the matrix of absolute stability is given by
R(Z1, Z2) = Im + ( ˜w T ⊗ Z1 + w T ⊗ Z2)(Iνm − Ã ⊗ Z1 − A ⊗ Z2) −1 e ⊗ Im.
At variance with the standard case, the spectral radius of the matrix does not depend only
on the eigenvalues of the matrices Z1 and Z2, since the two matrices B1 and B2 in general
do not have a common set of eigenvectors, they can not be diagonalized simultaneously.
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