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

Meenakshi Krishnan, K R Arun, Saurav Samantaray

Abstract:

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