# A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation

## TL;DR

This paper develops a unified IMEX Runge-Kutta method for hyperbolic systems with multiscale relaxation, ensuring accurate asymptotic behavior across different regimes, validated by numerical examples.

## Contribution

A novel IMEX Runge-Kutta scheme that maintains asymptotic preservation for multiscale hyperbolic systems with relaxation.

## Key findings

- The method accurately captures the correct asymptotic limit regardless of scaling.
- Numerical tests confirm the theoretical asymptotic preservation.
- The approach improves efficiency over standard methods in stiff regimes.

## Abstract

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation loose their efficiency and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable to capture the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis.