An efficient eigenvalue bounding method: CFL condition revisited

TL;DR

The paper tackles the time-step bottleneck in explicit CFD by deriving accurate eigenbound estimates for the discrete convective and diffusive NS operators without reconstructing time-dependent matrices. It introduces AlgEigCD, an inexpensive approach based on Gershgorin circle bounds, incidence-based operator reformulations, and precomputed matrices to bound the spectral radii, enabling larger stable time-steps when paired with a self-adaptive $\kappa 1L2$ scheme. The method demonstrates superior eigenbounds compared to classical CFL limits, especially on unstructured meshes, and yields substantial practical gains in $\Delta t$ in DNS/LES-type simulations. Its minimal, algebraic kernel footprint enhances cross-platform portability and efficiency, with potential extensions to other time-integration schemes and blending strategies.

Abstract

Direct and large-eddy simulations of turbulence are often solved using explicit temporal schemes. However, this imposes very small time-steps because the eigenvalues of the (linearized) dynamical system, re-scaled by the time-step, must lie inside the stability region. In practice, fast and accurate estimations of the spectral radii of both the discrete convective and diffusive terms are therefore needed. This is virtually always done using the so-called CFL condition. On the other hand, the large heterogeneity and complexity of modern supercomputing systems are nowadays hindering the efficient cross-platform portability of CFD codes. In this regard, our leitmotiv reads: relying on a minimal set of (algebraic) kernels is crucial for code portability and maintenance! In this context, this work focuses on the computation of eigenbounds for the above-mentioned convective and diffusive matrices which are needed to determine the time-step à la CFL. To do so, a new inexpensive method, that does not require to re-construct these time-dependent matrices, is proposed and tested. It just relies on a sparse-matrix vector product where only vectors change on time. Hence, both implementation in existing codes and cross-platform portability are straightforward. The effectiveness and robustness of the method are demonstrated for different test cases on both structured Cartesian and unstructured meshes. Finally, the method is combined with a self-adaptive temporal scheme, leading to significantly larger time-steps compared with other more conventional CFL-based approaches.

Figures (12)

• Figure 1: Left: face normal and neighbor labeling criterion. Right: definition of the volumes, $\mathsf{\Omega}_s$, associated with the the face-normal velocities, $\boldsymbol{u}_s$. Thick dashed rectangle is the volume associated with the staggered velocity $\mathrm{U}_{4} = [ \boldsymbol{u}_s ]_{4}$, i.e.$[ \mathsf{\Omega}_s ]_{4,4} = A_{4} \delta_{4}$ where $A_{4}$ is the face area and $\delta_{4} = | \boldsymbol{n}_{4} \cdot \overrightarrow{{c1} {c2}} |$ is the projected distance between adjacent cell centers. Thin dash-dotted lines are placed to illustrate that the sum of volumes is exactly preserved $\mathop{\mathrm{tr}}(\mathsf{\Omega}_s)=\mathop{\mathrm{tr}}(\mathsf{\Omega})= d \mathop{\mathrm{tr}}(\mathsf{\Omega}_c)$ ($d=2$ for 2D and $d=3$ for 3D) regardless of the location of the cell nodes.
• Figure 2: Stability region of the first-order forward Euler scheme (Eq. \ref{['forward_Euler']}) together with the family of $\kappa$-dependent second-order $\kappa$1L2 time-integration scheme (see Eqs. \ref{['k1L2']} and \ref{['off-step']} and \ref{['SAT_summary']}) (top) and their envelope (bottom). The shaded region is a graphical representation of the Bendixson theorem (see Theorem \ref{['Bendixson_theorem']}).
• Figure 3: Two-dimensional air-filled ($Pr=0.71$) differentially heated cavity at $Ra=10^{9}$ in a square domain. Left: schema of the flow configuration together with a flow visualization of the temperature field corresponding to the statistically steady state. Right: unstructured mesh used for the present tests. It is composed of $565$ triangular elements stretched to the walls.
• Figure 4: Numerical results obtained for the two-dimensional air-filled differentially heated cavity displayed in Figure \ref{['schema_DHC']} using a Cartesian stretched mesh with $23 \times 23 = 529$ control volumes (top) and an unstructured mesh composed of $565$ triangular elements (bottom). Notice that for this particular case, i.e. estimation of the spectral radius of $\mathsf{\Omega}_c^{-1} \mathsf{C}\left( \boldsymbol{u}_s \right)$ with a second-order symmetry-preserving discretization, the estimations of the EigenCD method are exactly the same as those given by the discretization-agnostic approach given in Eq.(\ref{['CFL_ANSYS-Fluent']}).
• Figure 5: Schema of the Rayleigh-Bénard configuration studied displayed together with an instantaneous temperature field corresponding to the air-filled ($Pr=0.7$) DNS (mesh RBC1e10-MeshA in Table \ref{['meshes_RBC']}) at $Ra=10^{10}$DABTRI15-TOPO-RBDABTRI19-3DTOPO-RB.

Theorems & Definitions (14)

• Theorem 1: Bendixson BEN1902
• Theorem 2
• proof
• Remark 1
• Theorem 3: Perron--Frobenius theorem PER1907FRO1912
• Theorem 4: Wielandt's theorem GRA07
• Theorem 5: Lemma 2 in Nikiforov NIK07
• Remark 2
• Theorem 6
• proof