Sparse Data Structures for Efficient State-to-State Kinetic Simulations

TL;DR

A more efficient way to represent the Jacobian arising in first-order implicit simulations for compressible flow physics coupled with StS models is introduced, consisting of a fully-sparse matrix and block-wise rank-one updates, whose overall complexity grows linearly with the number of quantum levels.

Abstract

Higher-fidelity entry simulations can be enabled by integrating finer thermo-chemistry models into compressible flow physics. One such class of models are State-to-State (StS) kinetics, which explicitly track species populations among quantum energy levels. StS models can represent thermo-chemical non-equilibrium effects that are hardly captured by standard multi-temperature models. However, the associated increase in computational cost is dramatic. For implicit solution techniques that rely on standard block-sparse representations of the Jacobian, both the spatial complexity and the temporal complexity grow quadratically with respect to the number of quantum levels represented. We introduce a more efficient way to represent the Jacobian arising in first-order implicit simulations for compressible flow physics coupled with StS models. The key idea is to recognize that the density of local blocks of the Jacobian comes from rank-one updates that can be managed separately. This leads to a new Jacobian structure, consisting of a fully-sparse matrix and block-wise rank-one updates, whose overall complexity grows linearly with the number of quantum levels. This structure also brings forth a potentially faster variation of the block-Jacobi preconditioning algorithm by leveraging the Sherman-Morrison-Woodbury inversion formula.

Figures (8)

• Figure 1: First $2000 \times 2000$ block of the discrete Jacobian associated with the implicit scheme on a sample $16 \times 32$ structured grid for varying values of $m$ (number of governing equations).
• Figure 2: Sparsity patterns of the temporal and normal flux Jacobians for $m = 50$.
• Figure 3: r1-sparse representation of the local trace flux Jacobian $\bf{\hat{A}}$ for $m = 50$.
• Figure 4: Sparsity patterns of temporal source contributions for $m=50$ ($46$ energy levels of $N_{2}$). The bandwidth depends on the threshold energy jump $\Delta v$ of the model.
• Figure 5: r1-sparse representation of the local temporal source Jacobian $\bf{G}$ for $m = 50$.