**Shi Jin**

*Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review*

**Pages:** 177-216

**Received:** 15 February 2011

**Accepted:** 31 March 2011

**Mathematics Subject Classification (2010):** 35L02, 82C70, 65M06.

**Keywords:**Kinetic equations, hyperbolic equations with relaxation, fluid dynamic limit, asymptotic-preserving schemes, quasi-neutral limit.

**Abstract:**
Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or
reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical
approximations become prohibitively expensive. Asymptotic-preserving (AP) schemes are schemes that are
efficient in these asymptotic regimes. The designing principle of AP schemes are to preserve,
*at the discrete level*, the asymptotic limit that drives one (usually the microscopic) equation to its
asymptotic (macroscopic) equation. An AP scheme is based on solving the microscopic equation,
instead of using a multiphysics approach that couples different physical laws at different scales.
When the small scale is not numerically resolved, an AP scheme *automatically* becomes a macroscopic
solver for the limiting equation. The AP methodology offers simple, robust and efficient computational
methods for a large class of multiscale kinetic, hyperbolic and other physical problems.
This paper reviews the basic concept, designing principle and some representative AP schemes.

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