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Half-explicit Runge-Kutta integrators for variational multiscale turbulence modeling: Toward higher-order accuracy in space and time

Yujie Sun, Chi Ding, Ju Liu

TL;DR

The paper develops a mathematically consistent framework that integrates half-explicit Runge-Kutta time integration with residual-based variational multiscale turbulence modeling via the Rothe method. This yields a Darcy-type coarse-scale problem at each RK stage and a bubble-enriched, subgrid-scale closure with a stabilization parameter independent of mesh size. Fourier analysis confirms enhanced stability, dispersion, and dissipation properties relative to conventional vertical approaches, and numerical tests (Taylor-Green vortex and open cavity flow) demonstrate superior kinetic-energy evolution, energy spectra, and limit-cycle fidelity. Overall, the approach delivers higher-order temporal accuracy, robust performance in convection-dominated regimes, and improved LES fidelity for complex turbulent flows, with efficient linear solves enabled by structured block matrices and preconditioning.

Abstract

The residual-based variational multiscale (VMS) formulation has achieved remarkable success in large-eddy simulation of turbulent flows. However, its temporal discretization has largely remained limited to second-order implicit schemes. The present work aims at advancing this direction through the introduction of Runge-Kutta (RK) schemes within the VMS framework in a mathematically consistent manner. Guided by the Rothe method, the half-explicit RK scheme is employed as its accuracy is theoretically guaranteed for index-2 differential-algebraic equations. Owing to the explicit treatment of the nonlinear term, the resulting spatial problem exhibits a structure analogous to that of the Darcy equation. Following the philosophy of the VMS analysis, a subgrid-scale model is derived without invoking linearization based on perturbation series and related assumptions. The analysis further reveals that the parameter in the subgrid model is independent of the spatial mesh size. Fourier analysis demonstrates that the Rothe method, compared with the conventional vertical method of lines, provides improved dissipation and dispersion properties and exhibits a larger stability region for convection-dominated regimes. In the Taylor-Green vortex benchmark, the proposed schemes demonstrate superior performance as a large-eddy simulation model, achieving higher fidelity in predicting the kinetic energy evolution, energy spectra, and vortex structures than the conventional VMS formulation. Simulations of the open cavity flow further show that the proposed schemes can accurately capture the periodic limit cycle caused by the supercritical Hopf bifurcation, confirming its effectiveness and fidelity for highly sensitive flow instability problems.

Half-explicit Runge-Kutta integrators for variational multiscale turbulence modeling: Toward higher-order accuracy in space and time

TL;DR

The paper develops a mathematically consistent framework that integrates half-explicit Runge-Kutta time integration with residual-based variational multiscale turbulence modeling via the Rothe method. This yields a Darcy-type coarse-scale problem at each RK stage and a bubble-enriched, subgrid-scale closure with a stabilization parameter independent of mesh size. Fourier analysis confirms enhanced stability, dispersion, and dissipation properties relative to conventional vertical approaches, and numerical tests (Taylor-Green vortex and open cavity flow) demonstrate superior kinetic-energy evolution, energy spectra, and limit-cycle fidelity. Overall, the approach delivers higher-order temporal accuracy, robust performance in convection-dominated regimes, and improved LES fidelity for complex turbulent flows, with efficient linear solves enabled by structured block matrices and preconditioning.

Abstract

The residual-based variational multiscale (VMS) formulation has achieved remarkable success in large-eddy simulation of turbulent flows. However, its temporal discretization has largely remained limited to second-order implicit schemes. The present work aims at advancing this direction through the introduction of Runge-Kutta (RK) schemes within the VMS framework in a mathematically consistent manner. Guided by the Rothe method, the half-explicit RK scheme is employed as its accuracy is theoretically guaranteed for index-2 differential-algebraic equations. Owing to the explicit treatment of the nonlinear term, the resulting spatial problem exhibits a structure analogous to that of the Darcy equation. Following the philosophy of the VMS analysis, a subgrid-scale model is derived without invoking linearization based on perturbation series and related assumptions. The analysis further reveals that the parameter in the subgrid model is independent of the spatial mesh size. Fourier analysis demonstrates that the Rothe method, compared with the conventional vertical method of lines, provides improved dissipation and dispersion properties and exhibits a larger stability region for convection-dominated regimes. In the Taylor-Green vortex benchmark, the proposed schemes demonstrate superior performance as a large-eddy simulation model, achieving higher fidelity in predicting the kinetic energy evolution, energy spectra, and vortex structures than the conventional VMS formulation. Simulations of the open cavity flow further show that the proposed schemes can accurately capture the periodic limit cycle caused by the supercritical Hopf bifurcation, confirming its effectiveness and fidelity for highly sensitive flow instability problems.

Paper Structure

This paper contains 35 sections, 1 theorem, 68 equations, 19 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The Rothe method
  3. The RK methods for the incompressible NS equations
  4. The VMS formulation and the Darcy problem
  5. Strong- and weak-form problems
  6. Half-explicit Runge-Kutta methods
  7. Conceptural framework: HERK method for an index-2 DAE system
  8. HERK method for the weak-form problem
  9. Variational multiscale formulation
  10. Multiscale decomposition
  11. Approximation of the projector via bubble enrichment
  12. Fourier analysis
  13. Fully discrete schemes
  14. RK$(s,p)$-VMS
  15. VMS-RK$(s,p)$
  16. ...and 20 more sections

Key Result

Proposition 1

If $\mathcal{G}_{,\bm y}\mathcal{F}_{,\bm z}$ is non-singular, the consistency of the initial condition is satisfied, the local error of the scheme is of order $p+1$, the coefficient $b_s \neq 0$, and $a_{i\: i-1} \neq 0$ for $i=2, \cdots, s$, the method has a convergence order of $p$, that is, $\bm

Figures (19)

  • Figure 1: Stability regions for RK$(s,p)$-VMS and VMS-RK$(s,p)$ schemes.
  • Figure 2: Algorithmic damping (top) and frequency ratios (bottom) for schemes with $s=p=1$.
  • Figure 3: Algorithmic damping (top) and frequency ratios (bottom) for schemes with $s=p=2$.
  • Figure 4: Algorithmic damping (top) and frequency ratios (bottom) for schemes with $s=p=3$.
  • Figure 5: Algorithmic damping (top) and frequency ratios (bottom) for schemes with $s=p=4$.
  • ...and 14 more figures

Theorems & Definitions (7)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6