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

Stability analysis
When studying the A-stability of a RK scheme, one considers a scalar equation of the
form
y ′ = λy, y(0) = 1
with y : R + → R, and λ ∈ R, ℜλ ≤ 0. Such test problem is sufficient to characterize the
stability property of a Runge-Kutta scheme when applied to a linear system of the form
with y ∈ R m , and B ∈ R m×m .
y ′ = By, y(0) = y0
In the case of IMEX-RK, and in general for additive Runge-Kutta methods, such a stability
analysis has a limited validity. In fact, consider a generic linear system of the form
y ′ = B1 y + B2 y
with y ∈ R m , and B1, B2 ∈ R m×m , and apply an IMEX scheme which is explicit in B1y and
implicit in B2y. The stability of the numerical solution depends on the two matrices B1
and B2, and not only on their eigenvalues since in general the two matrices do not share
the same eigenvectors, and therefore they can not be diagonalized simultaneously.
18