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

Remark
• Clearly one may claim that if the implicit part of the IMEX scheme is A-stable or
L-stable the previous theorem is satisfied. Note however that this is true only if the
tableau of the implicit integrator does not contain any column of zeros that makes
it reducible to a simpler A-stable or L-stable form.
• Finally we observe that this result does not guarantee the accuracy of the solution
for the N − n non conserved quantities. In fact, since the very last step in the scheme
it is not a projection towards the local equilibrium, a final layer effect occurs.
It is easy to show that
Corollary 1 If det A �= 0 and wj = aνj, j = 1, . . . ν then in the limit ɛ → 0, the IMEX
scheme is asymptotically accurate, that is it provides the order of accuracy of the explicit
RK scheme characterized by (Ã, ˜w, ˜c) for both conserved and non conserved variables
We recall that the additional condition wj = aνj, j = 1, . . . ν makes an A−stable method
L−stable. Usually these methods are referred to as stiffly accurate.
15