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An all Mach number semi-implicit hybrid Finite Volume/Virtual Element method for compressible viscous flows on Voronoi meshes

Walter Boscheri, Saray Busto, Michael Dumbser

TL;DR

The paper develops a high-order all-Mach number solver for compressible viscous flows on Voronoi meshes by splitting the Navier–Stokes system into an explicit convective sub-system and implicit viscous/pressure sub-systems discretized with a novel virtual element method. High temporal and spatial accuracy is achieved via IMEX Runge–Kutta time stepping and CWENO reconstructions, with transfer operators ensuring consistent FV–VEM coupling on general polygons. The scheme is asymptotic preserving in both the incompressible and low-Reynolds-number limits and globally energy-conserving through a dedicated second pressure solve. Extensive numerical tests across subsonic to hypersonic regimes validate accuracy, robustness, and the AP property, while leveraging polygonal grids for complex geometries. The work offers a flexible, robust framework for multi-scale compressible flows with potential extensions to MHD and moving meshes.

Abstract

We present a novel high order semi-implicit hybrid finite volume/virtual element numerical scheme for the solution of compressible flows on Voronoi tessellations. The method relies on the flux splitting of the compressible Navier-Stokes equations into three sub-systems: a convective sub-system solved explicitly using a finite volume (FV) scheme, and the viscous and pressure sub-systems which are discretized implicitly at the aid of a virtual element method (VEM). Consequently, the time step restriction of the overall algorithm depends only on the mean flow velocity and not on the fast pressure waves nor on the viscous eigenvalues. As such, the proposed methodology is well suited for the solution of low Mach number flows at all Reynolds numbers. Moreover, the scheme is proven to be globally energy conserving so that shock capturing properties are retrieved in high Mach number flows. To reach high order of accuracy in time and space, an IMEX Runge-Kutta time stepping strategy is employed together with high order spatial reconstructions in terms of CWENO polynomials and virtual element space basis functions. The chosen discretization techniques allow the use of general polygonal grids, a useful tool when dealing with complex domain configurations. The new scheme is carefully validated in both the incompressible limit and the high Mach number regime through a large set of classical benchmarks for fluid dynamics, assessing robustness and accuracy.

An all Mach number semi-implicit hybrid Finite Volume/Virtual Element method for compressible viscous flows on Voronoi meshes

TL;DR

The paper develops a high-order all-Mach number solver for compressible viscous flows on Voronoi meshes by splitting the Navier–Stokes system into an explicit convective sub-system and implicit viscous/pressure sub-systems discretized with a novel virtual element method. High temporal and spatial accuracy is achieved via IMEX Runge–Kutta time stepping and CWENO reconstructions, with transfer operators ensuring consistent FV–VEM coupling on general polygons. The scheme is asymptotic preserving in both the incompressible and low-Reynolds-number limits and globally energy-conserving through a dedicated second pressure solve. Extensive numerical tests across subsonic to hypersonic regimes validate accuracy, robustness, and the AP property, while leveraging polygonal grids for complex geometries. The work offers a flexible, robust framework for multi-scale compressible flows with potential extensions to MHD and moving meshes.

Abstract

We present a novel high order semi-implicit hybrid finite volume/virtual element numerical scheme for the solution of compressible flows on Voronoi tessellations. The method relies on the flux splitting of the compressible Navier-Stokes equations into three sub-systems: a convective sub-system solved explicitly using a finite volume (FV) scheme, and the viscous and pressure sub-systems which are discretized implicitly at the aid of a virtual element method (VEM). Consequently, the time step restriction of the overall algorithm depends only on the mean flow velocity and not on the fast pressure waves nor on the viscous eigenvalues. As such, the proposed methodology is well suited for the solution of low Mach number flows at all Reynolds numbers. Moreover, the scheme is proven to be globally energy conserving so that shock capturing properties are retrieved in high Mach number flows. To reach high order of accuracy in time and space, an IMEX Runge-Kutta time stepping strategy is employed together with high order spatial reconstructions in terms of CWENO polynomials and virtual element space basis functions. The chosen discretization techniques allow the use of general polygonal grids, a useful tool when dealing with complex domain configurations. The new scheme is carefully validated in both the incompressible limit and the high Mach number regime through a large set of classical benchmarks for fluid dynamics, assessing robustness and accuracy.
Paper Structure (33 sections, 1 theorem, 114 equations, 12 figures, 4 tables)

This paper contains 33 sections, 1 theorem, 114 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Numerical method
  4. Semi-discrete semi-implicit scheme
  5. Overall methodology
  6. High order extension in time: semi-implicit IMEX scheme
  7. General unstructured mesh
  8. Convective sub-system: Finite Volume scheme
  9. High order extension in space: CWENO reconstruction
  10. Viscous and pressure sub-systems: Virtual Element Method
  11. Variational formulation of the viscous sub-system
  12. Variational formulation of the pressure sub-system
  13. The virtual element space
  14. Elliptic projection operator
  15. $L_{2}$ projection operator
  16. ...and 18 more sections

Key Result

Theorem 3.1

Assuming impermeable boundary conditions $\int\limits_{\partial \Omega} \mathbf{w} \cdot \mathbf{n} = 0$, the semi-discrete scheme eqn.pressure_vf4 with the intermediate results for pressure and kinetic energy, given by eqn.pressure_tdwv and eqn.pressure_tdsv, respectively, is globally energy conser

Figures (12)

  • Figure 1: Riemann problems. RP1 at time $t_f=0.2$ (top) and RP2 at time $t_f=0.15$. Left: density $\rho$. Center: horizontal velocity component $u$. Right: pressure $p$.
  • Figure 2: Riemann problems. RP3 at time $t_f=0.14$ (top) and RP4 at time $t_f=0.012$. Left: density $\rho$. Center: horizontal velocity component $u$. Right: pressure $p$.
  • Figure 3: Riemann problems. Time evolution of the total energy for the globally energy conservative (black line) and the non-conservative (red line) SI-FVVEM schemes for RP1 (left) and RP4 (right).
  • Figure 4: Circular explosion problem at time $t_f=0.25$. Top left: two-dimensional view of the density distribution. From top-left to bottom-right: density $\rho$, horizontal velocity $u$ and pressure $p$ distribution compared against the reference solution extracted with a one-dimensional cut of 200 equidistant points along the $x-$direction at $y=0$.
  • Figure 5: Double Mach reflection problem at time $t_f=0.2$. Top: 21 equidistant contour lines in the range $[50,500]$ for pressure. Bottom: zoom on the shock front with density (left) and vorticity (right) distribution.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Remark 3.1: Asymptotic preserving property
  • Remark 3.2: Numerical integration
  • Theorem 3.1
  • proof