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

Asymptotic properties of IMEX schemes
An IMEX scheme for an hyperbolic system with relaxation has the form
U (i)
i = U0
�i−1
+ h
U1 = U0 + h
j=1
ν�
i=1
ãijF (U (j) ) + h
˜wiF (U (i) ) + h
ν�
j=1
ν�
i=1
1
aij
ε R(U (j) ),
1
wi
ε R(U (i) ).
Definition 1 We say that an IMEX scheme for an hyperbolic system with relaxation
is asymptotic preserving (AP) if in the limit ɛ → 0 the scheme becomes a consistent
discretization of the limit system of conservation laws. We use the notation APk if the
scheme is of order k in the limit ɛ → 0.
Note that this definition does not imply that the scheme preserves the order of accuracy
in t in the stiff limit ɛ → 0. In the latter case the scheme is said asymptotically accurate.
Examples: Scheme SP(1,1,1) is clearly AP1. Scheme Jin(2,2,2) is AP2, but it is not
uniformly valid in ε. Schemes Midpoint(1,2,2) and CN(2,2,2) are not AP even if both
implicit parts of the schemes are A-stable. On the contrary, schemes CJR(3,2,2) and
LRR(3,2,2) are AP and uniformly valid in ε, but only scheme LRR(3,2,2) is AP2.
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