the Boussinesq approximation is constant, while in the anelastic approximation is a known function of height z.
This paper presents a new forward-in-time advection method for nearly incompressible flow, MU, and its application to an adaptive For many flows, a ''multilevel'' scheme that incorporates multilevel flow solver for atmospheric flows. MU is a modification regions of higher grid refinement in time and space can of Leonard et al.'s UTOPIA scheme. MU, like UTOPIA, is based on be advantageous to track localized fine-scale flow features.
third-order accurate semi-Lagrangian multidimensional upwinding
Advection schemes should ideally be efficient, highly for constant velocity flows. For varying velocity fields, MU is a second-order conservative method. MU has greater stability and accurate, preserve in discrete form the conservation laws accuracy than UTOPIA and naturally decomposes into a monotone governing the advected quantities, and be ''monotone'' low-order method and a higher-order accurate correction for use (not introduce spurious overshoots and undershoots). Difwith flux limiting. Its stability and accuracy make it a computationferent types of advection schemes have evolved in response ally efficient alternative to current finite-difference advection methto these requirements. Centered ''leapfrog'' differencing ods. We present a fully second-order accurate flow solver for the anelastic equations, a prototypical low Mach number flow. The flow in space and time, long popular for atmospheric applicasolver is based on MU which is used for both momentum and scalar tions, is simple and second-order accurate, but it is prone transport equations. This flow solver can also be implemented with to large numerical overshoots and must be time-filtered any forward-in-time advection scheme. The multilevel flow solver for stability. Semi-Lagrangian methods [30,32] have also conserves discrete global integrals of advected quantities and includes adaptive mesh refinement. Its second-order accuracy is veri-been proven attractive for scalar advection, primarily due fied using a nonlinear energy conservation integral for the anelastic to their stability and accuracy properties. However, they equations. For a typical geophysical problem in which the flow is are not conservative. 16, most rapidly varying in a small part of the domain, the multilevel 17, 24, 29] which use only the most recent time level to flow solver achieves global accuracy comparable to a uniform-resoadvance to a new time level are more complex than leaplution simulation for 10% of the computational cost.