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The design and analysis of a unified asymptotic preserving (AP) andwell-balanced scheme for the Euler Equations with gravitational and frictionalsource terms is presented in this paper. The asymptotic behaviour of the Eulersystem in the limit of zero Mach and Froude numbers, and large friction ischaracterised by an additional scaling parameter. Depending on the values ofthis parameter, the Euler system relaxes towards a hyperbolic or a paraboliclimit equation. Standard Implicit-Explicit Runge-Kutta schemes are incapable ofswitching between these asymptotic regimes. We propose a timesemi-discretisation to obtain a unified scheme which is AP for the twodifferent limits. A further reformulation of the semi-implicit scheme can berecast as a fully-explicit method in which the mass update contains bothhyperbolic and parabolic fluxes. A space-time fully-discrete scheme is derivedusing a finite volume framework. A hydrostatic reconstruction strategy, anupwinding of the sources at the interfaces, and a careful choice of the centraldiscretisation of the parabolic fluxes are used to achieve the well-balancingproperty for hydrostatic steady states. Results of several numerical casestudies are presented to substantiate the theoretical claims and to verify therobustness of the scheme.

A Unified Asymptotic Preserving and Well-balanced Scheme for the Euler System with Multiscale Relaxation