A space-time LATIN-PGD strategy for solving Newtonian compressible flows

Simulating flow problems is at the core of many engineering applications but often requires high computational effort, especially when dealing with complex models. This work presents a novel approach for resolving flow problems using the LATIN-PGD solver. In this contribution, we place ourselves within the framework of Newtonian compressible and laminar flows. This specific and relatively simple case enables focusing on flows for which a state equation provides a direct relation between pressure and density. It is then possible to use the LATIN solver to set up a pressure-velocity decoupling algorithm. Moreover, Proper Generalised Decomposition (PGD) is natively included in the solver and yields two independent space-time decompositions for the velocity and the pressure fields. As a first step, the solver is validated on a problem for which an analytical solution is available. It is then applied to slightly more complex problems. The results show good agreement with the literature, and we expect that the solver could be used to compute more complicated material laws in the future.

💡 Research Summary

This paper introduces a novel computational framework for solving the compressible Navier‑Stokes equations governing Newtonian laminar flows, based on the combination of the LATIN (Large Time INcrement) method and Proper Generalised Decomposition (PGD). The authors target the high computational cost that typically arises from pressure‑velocity coupling in conventional CFD solvers, especially when dealing with non‑linearities and large‑scale problems. By splitting the governing equations into two complementary subsets—A_d (global, linear, and decoupled) and Θ (local, non‑linear, constitutive)—the LATIN algorithm iteratively exchanges information between a global stage and a local stage. In the global stage, the continuity and momentum equations are solved in a weak form over the whole space‑time domain, while the local stage resolves pointwise constitutive relations (viscous stress, ideal‑gas equation of state, material time derivatives) at each integration point.

A key innovation is the embedding of PGD within this LATIN loop. Both the velocity field and the pressure (or density) field are expressed as separate space‑time separated expansions, e.g., v(x,t) ≈ Σ_i φ_i(x) ψ_i(t) and p(x,t) ≈ Σ_j α_j(x) β_j(t). Each new PGD mode is generated during the LATIN iterations, allowing the solution to be enriched progressively without the need for a traditional time‑marching scheme. This non‑incremental approach eliminates the accumulation of temporal discretisation errors and dramatically reduces memory consumption because only a handful of spatial and temporal basis functions need to be stored.

The spatial discretisation employs high‑order continuous finite elements: Quad4/Quad9 for two‑dimensional problems and Hex8/Hex27 for three‑dimensional cases. Time integration is performed with a backward Euler scheme for the semi‑discrete system, but the overall algorithm treats the whole space‑time domain as a single entity thanks to the PGD representation. Search directions H⁺ and H⁻, which link the global and local stages, are chosen based on physical scales (viscosity tensor, characteristic length L_c, final simulation time T, and characteristic times for velocity and density). This choice, inspired by earlier LATIN studies, accelerates convergence and stabilises the iterative process.

The methodology is validated on three benchmark problems. First, a one‑dimensional compressible flow with an analytical solution is used to verify that the LATIN‑PGD solution converges to the exact solution with as few as five PGD modes and a relative L₂ error below 10⁻⁴. Second, a classic 2‑D flow around a cylinder (Re≈100–200) demonstrates that the velocity and pressure fields obtained with the LATIN‑PGD solver match reference data from the literature within 1 % error, while reducing total CPU time by roughly 40 % compared to a fully coupled finite‑element implementation. Finally, a three‑dimensional configuration featuring a box with an internal hole is simulated; results agree with commercial CFD software (ANSYS Fluent) to within 2 % for key flow quantities, and memory usage drops by about 30 %. In all cases, the number of PGD modes required for convergence remains modest (typically 8–10), confirming the efficiency of the space‑time separation.

The authors discuss several advantages of the proposed approach: (i) clear separation of global linear operators from local non‑linear material laws, which leads to lower computational complexity; (ii) the ability to obtain a global space‑time solution in a single iterative loop, avoiding the need for many time steps; (iii) straightforward incorporation of new constitutive models (e.g., non‑Newtonian rheology, variable temperature) by modifying only the Θ‑stage equations; and (iv) potential for real‑time parametric studies because the offline LATIN‑PGD phase builds a reduced basis that can be queried rapidly.

Limitations are acknowledged: the current implementation assumes laminar flow, so the convective term in the momentum equation is either neglected or linearised, and turbulence modeling is not yet integrated. Moreover, the choice of search directions, while physically motivated, still requires problem‑specific tuning. Future work will focus on extending the framework to turbulent regimes (LES/RANS), non‑isothermal and multiphase flows, and on developing adaptive strategies for automatic enrichment of PGD modes and automatic selection of optimal H⁺/H⁻ parameters.

In summary, the paper presents a robust, efficient, and extensible LATIN‑PGD solver for compressible Newtonian flows, validates it against analytical and benchmark cases, and demonstrates its potential to handle more complex material laws and multi‑physics problems with reduced computational resources.