On the Reynolds-number scaling of Poisson solver complexity

TL;DR

The paper investigates how the computational cost of solving the pressure Poisson equation in DNS of incompressible flows scales with Reynolds number. By combining physical scaling arguments from turbulence theory with a detailed analysis of Jacobi and multigrid convergence, the authors derive power-law relationships for the residual and establish a Reynolds-number phase space that predicts the trend of solver iterations. They validate the theory across homogeneous isotropic turbulence, Rayleigh-Bénard convection, bluff-body wakes, and a 1D Burgers model, showing NS turbulence tends to reduce iteration counts with increasing Re while Burgers shows the opposite. The framework yields explicit expressions for the convergence exponent xi and demonstrates that multigrid maintains the same Re-scaling while accelerating convergence, offering guidance for next-generation preconditioning for extreme-scale CFD.

Abstract

We aim to answer the following question: is the complexity of numerically solving the Poisson equation increasing or decreasing for very large simulations of incompressible flows? Physical and numerical arguments are combined to derive power-law scalings at very high Reynolds numbers. A theoretical convergence analysis for both Jacobi and multigrid solvers defines a two-dimensional phase space divided into two regions depending on whether the number of solver iterations tends to decrease or increase with the Reynolds number. Numerical results indicate that, for Navier-Stokes turbulence, the complexity decreases with increasing Reynolds number, whereas for the one-dimensional Burgers equation it follows the opposite trend. The proposed theoretical framework thus provides a unified perspective on how solver convergence scales with the Reynolds number and offers valuable guidance for the development of next-generation preconditioning and multigrid strategies for extreme-scale simulations.

Figures (14)

• Figure 1: Illustrative explanation of the two competing effects on the solution of Poisson's equation when increasing $Re$ number: time-step, $\Delta t$, decreases leading to smaller values of the initial residual, $\hat{r}_{k}^{0}$, whereas the range of scales increases.
• Figure 2: Phase space $\{ \tilde{\alpha} , \tilde{\beta} \}$. Solid black line corresponds to $|| r^n ||^2 \propto Re^{0}$ in Eqs.(\ref{['Jacobi_residual_scaling']}) and (\ref{['MG_residual_scaling']}), i.e. neutral effect of $Re$-number in the total number of iterations, and corresponds to $\tilde{\alpha} = - \tilde{\beta}$. Horizontal blue line corresponds to $\tilde{\beta}=7/4$ which is the estimation for the NS equations. The blue dot labeled as NS corresponds to the most common situation where $q=2$ (see Eq. \ref{['residual3']}) and $\alpha=-3/4$ (see Eq. \ref{['Dt_scaling']}) leading to $\tilde{\alpha}=-5/2$ (see Eq. \ref{['alphatilde_def']}). The horizontal red line corresponds to the same analysis but for the Burgers' equation studied in Section \ref{['results']}.
• Figure 3: Energy and pressure spectra for the forced HIT simulation at $Re_\lambda \approx 433$. Data has been obtained from the JHTDB database PER07-JHTDBLI08-JHTDB.
• Figure 4: Same as in Figure \ref{['Energy_and_pressure_spectra']} but for the second, $Q_{\mathsf{G}}$, and third invariant, $R_{\mathsf{G}}$, of the velocity gradient tensor.
• Figure 5: Same as in Figure \ref{['Energy_and_pressure_spectra']} but for the convective term, $( \boldsymbol{u} \cdot \nabla) Q_{\mathsf{G}}$, and the residual of the Poisson equation.