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Computational study of numerical flux schemes for mesoscale atmospheric flows in a Finite Volume framework

Nicola Clinco, Michele Girfoglio, Annalisa Quaini, Gianluigi Rozza

TL;DR

This work develops a density-based Finite Volume solver for the mildly compressible, non-hydrostatic Euler equations governing dry mesoscale atmospheric flows. It achieves well-balanced behavior through a local hydrostatic reconstruction and evaluates four approximate Riemann solvers—Roe-Pike, HLLC, AUSM+-up, and HLLC-AUSM—via two classical benchmarks: the smooth rising thermal bubble and the density current. The results show that AUSM+-up and HLLC-AUSM are less dissipative and can operate on coarser meshes with good fidelity, with HLLC-AUSM offering the best agreement with literature in coarse-to-moderate grids, while Roe-Pike and HLLC are more diffusive on coarser meshes and differences fade with refinement. The findings provide practical guidance for selecting flux solvers in density-based FV schemes for efficient and accurate mesoscale atmospheric simulations. All mathematical notation is kept within $...$ delimiters where present.

Abstract

We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based approach and compare the accuracy of these four approximated Riemann solvers using two two classical benchmarks, namely the smooth rising thermal bubble and the density current.

Computational study of numerical flux schemes for mesoscale atmospheric flows in a Finite Volume framework

TL;DR

This work develops a density-based Finite Volume solver for the mildly compressible, non-hydrostatic Euler equations governing dry mesoscale atmospheric flows. It achieves well-balanced behavior through a local hydrostatic reconstruction and evaluates four approximate Riemann solvers—Roe-Pike, HLLC, AUSM+-up, and HLLC-AUSM—via two classical benchmarks: the smooth rising thermal bubble and the density current. The results show that AUSM+-up and HLLC-AUSM are less dissipative and can operate on coarser meshes with good fidelity, with HLLC-AUSM offering the best agreement with literature in coarse-to-moderate grids, while Roe-Pike and HLLC are more diffusive on coarser meshes and differences fade with refinement. The findings provide practical guidance for selecting flux solvers in density-based FV schemes for efficient and accurate mesoscale atmospheric simulations. All mathematical notation is kept within delimiters where present.

Abstract

We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based approach and compare the accuracy of these four approximated Riemann solvers using two two classical benchmarks, namely the smooth rising thermal bubble and the density current.
Paper Structure (14 sections, 61 equations, 7 figures, 2 tables)

This paper contains 14 sections, 61 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Space Discretization
  4. Treatment of the flux term
  5. The Roe-Pike method
  6. The Harten–Lax–van Leer contact method
  7. The AUSM$^{+}$-up method
  8. The HLLC-AUSM method
  9. Time discretization
  10. Numerical Results
  11. Hydrostatic atmosphere
  12. Smooth rising thermal bubble
  13. Density current
  14. Concluding remarks

Figures (7)

  • Figure 1: Close-up view of two orthogonal control volumes.
  • Figure 2: Close-up view of two non-orthogonal control volumes.
  • Figure 3: Hydrostatic atmosphere: time evolution of the maximal vertical velocity $w_{max}$ for all the methods for the computation of the numerical flux under consideration.
  • Figure 4: Rising thermal bubble: perturbation of the potential temperature computed at $t = 600$ s with mesh $h= 2.5$ m and the Roe-Pike (top left), HLLC (top right), AUSM$^{+}$-up (bottom letft), and HLLC-AUSM (bottom right) methods.
  • Figure 5: Rising thermal bubble: profile of the potential temperature perturbation along $z=700$ m at $t=600$ s given by the different methods to compute the numerical flux with mesh $h=2.5$ m (left) and mesh $h=5$ m (right). Reference 1 is taken from ResGir2009, while Reference 2 is from GEA_GIR_QUA.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Remark 1