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

Asymptotic behavior
Relaxation operators and zero-relaxation limit
Let us consider an hyperbolic system with relaxation
∂tU + ∂xF (U) = 1
R(U), x ∈ R.
ε
The operator R : R N → R N is said a relaxation operator if there exists a constant n × N
matrix Q with rank(Q) = n < N such that
QR(U) = 0 ∀ U ∈ R N .
This gives n independent conserved quantities u = QU that uniquely determine a local
equilibrium U = E(u), such that R(E(u)) = 0.
We obtain a system of n conservation laws which is satisfied by every solution of the
relaxation system
∂t(QU) + ∂x(QF (U)) = 0.
As ε → 0 we get R(U) = 0 which implies U = E(u). In this case the relaxation system is
well approximated by the reduced system
where G(u) = QF (E(u)).
∂tu + ∂xG(u) = 0,
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