Iterated upwind schemes for gas dynamics

Abstract

A class of high-resolution schemes established in integration of anelastic equations is extended to fully compressible flows, and documented for unsteady (and steady) problems through a span of Mach numbers from zero to supersonic. The schemes stem from iterated upwind technology of the multidimensional positive definite advection transport algorithm (MPDATA). The derived algorithms employ standard and modified forms of the equations of gas dynamics for conservation of mass, momentum and either total or internal energy as well as potential temperature. Numerical examples from elementary wave-propagation, through computational aerodynamics benchmarks, to atmospheric small-and large-amplitude acoustics with intricate wave-flow interactions verify the approach for both structured and unstructured meshes, and demonstrate its flexibility and robustness.

An unstructured-mesh atmospheric model for nonhydrostatic dynamics

“…The outlined formulation of the BVP dictated by a forward-in-time model readily extends to flow problems with Coriolis acceleration, Newtonian cooling and Rayleigh friction [22], whereas more cumbersome linear functionals and nonlinear terms can be delegated to the explicit counterpart R by means of outer iteration or its special case the predictor/corrector approach [9,36,38]. There is no single principle guiding the degree of the semi-implicitness in a model design, yet semi-implicit schemes are usually selected for computational stability, admitting large time step integrations.…”

“…Because the time scale of Coriolis acceleration (24h) is orders of magnitude larger than mesoscale δt ∼ O(10) s, the scheme converges rapidly, in a few outer iterations [36]. Additional solutions of the elliptic problems are cost-wise minuscule, because the work of the solver decreases dramatically beyond the first iteration; see [38] for a related discussion.…”

“…In a series of papers a class of the edge-based unstructured-mesh MPDATA schemes [32,33] is derived from first principles, and the corresponding unstructured mesh finite-volume NFT framework is formulated for modeling engineering [40,36] and atmospheric flows [41,42,38]. In particular, [41] introduces a class of global hydrostatic models that employ a classical geospherical reference frame with the governing equations cast in the latitude-longitude surfacebased coordinates, but resolve notorious issues associated with meridians convergence near the poles by exploiting the flexibility of unstructured mesh discretization.…”

An unstructured-mesh atmospheric model for nonhydrostatic dynamics: Towards optimal mesh resolution

“…13 already evince massive wave breaking with turbulent flow in the lee and evanescent perturbations, excited by convective eddies, penetrating through the critical layer. Notably, these results may be viewed as a 3D paraphrases of the 2D flow discussed in section 5 of [43]. At the first glance, the physical content of the two bottom panels of Fig.…”

“…The roots of the NFT schemes for unstructured meshes are in the derivation from first principles of the finitevolume multidimensional positive definite advection transport algorithm (MP-DATA) together with the stability and convexity theory, convergence analysis, and several MPDATA options [41,42]. 2 Further developments included an unstructured-mesh finite-volume NFT framework for gas dynamics with a selection of techniques for mesh adaptivity [50,43] originally developed for engineering flows [59,49,51]. The applicability of NFT methods to gas dynamics was demonstrated through verifications against benchmarks for all-speed aerodynamic flows with favourable comparisons to established solutions (theoretical and numerical) and convergence studies.…”