A unified asymptotic preserving and well-balanced scheme for the Euler system with multiscale relaxation

The design and analysis of a unified asymptotic preserving (AP) well-balanced scheme for the Euler Equations with gravitational frictional source terms is presented in this paper. behaviour system limit zero Mach Froude numbers, large friction characterised by an additional scaling parameter. Depending on values parameter, relaxes towards hyperbolic or parabolic equation. Standard Implicit–Explicit Runge–Kutta schemes are incapable switching between these regimes. We propose time semi-discretisation to obtain which AP two different limits. A further reformulation semi-implicit can be recast as fully-explicit method mass update contains both fluxes. space–time fully-discrete derived using finite volume framework. hydrostatic reconstruction strategy, upwinding sources at interfaces, careful choice central discretisation fluxes used achieve well-balancing property steady states. Results several numerical case studies substantiate theoretical claims verify robustness scheme.