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\centerline{\large \bf  Recent results on Runge-Kutta methods}

\centerline{\large \bf  for hyperbolic systems with relaxation}

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\centerline{Lorenzo Pareschi}
\centerline{\it University of Ferrara, Italy }
\centerline{e-mail l.pareschi@unife.it }
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We will present a unified approach of Runge-Kutta schemes for
hyperbolic systems with relaxation. These schemes consist of
applying an implicit discretization to the relaxation terms and an
explicit one for the flux. We show that most of the splitting
schemes can be written in the formalism of Implicit-Explicit
(IMEX) Runge-Kutta schemes, where the implicit solver is a
diagonally implicit (DIRK) scheme. Similarly it is easy to write
an IMEX Runge-Kutta scheme in splitting form. In particular, we
derive general conditions that guarantee the asymptotic preserving
property, i.e. the consistency of the scheme with the equilibrium
system, and show that the implicit step can be solved, in many
cases, every time we use a DIRK scheme. Accuracy, stability and
TVD properties of these schemes are studied both analytically and
numerically.

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