Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit

Sebastiano Boscarino, Lorenzo Pareschi, Giovanni Russo
(29/07/2011 SIAM J. Sci. Comp. 35 (2013), 22-51. arXiv:1110.4375)

We consider Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic and kinetic equations in the diffusion limit. In such regime the system relaxes towards a parabolic convection-diffusion equation and it is desirable to have a method that is able to capture the asymptotic behavior with an implicit treatment of the limiting diffusive terms.
To this goal we reformulate the problem by properly combining the limiting diffusion flux with the convective flux. This, however, introduces new difficulties due to the dependence of the stiff source term on the gradient. Thus, by an accurate analysis of the different type of IMEX schemes, we proposed several schemes that under some assumptions show good behavior with respect to the small scaling parameter in the zero relaxation limit. In particular, at variance with the classical fluid-limit, our approach originates in the zero relaxation limit an IMEX method for the corresponding convection-diffusion system. Several numerical examples including neutron transport equations confirm the theoretical analysis.

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