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

Example:
A simple prototype example of relaxation system is given by
∂tu + ∂xf1(u, v) = 0,
∂tv + ∂xf2(u, v) = − 1
(v − e(u)),
ε
which corresponds to U = (u, v), F (U) = (f1(u, v), f2(u, v)), R(U) = (0, e(u) − v).
As ε → 0 we get the local equilibrium v = e(u) and setting G(u) = f1(u, e(u)) the reduced
system of conservation laws
Numerical requirements
∂tu + ∂xG(u) = 0.
• In most cases F (U) is non stiff and 1
R(U) contains the stiffness. It is desirable to
ε
develop numerical schemes which are explicit in F and implicit in R.
• It is essential that the numerical scheme is accurate for the reduced limit system of
conservation laws. This property is related to L-stability.
• The schemes should be high resolution shock capturing, yielding correct shock location
and speed without numerical oscillations.
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