A memory-efficient deterministic method for multiscale gas flows using an ensemble-of-subproblems strategy with stochastic discrete velocities

Abstract

Deterministic multiscale gas flow simulations have long suffered from the curse of dimensionality: the number of discrete velocities increases dramatically with the velocity space dimension and the Mach number, exhausting available memory and computational resources. To address this issue, this paper proposes a memory-efficient deterministic method based on an ensemble-of-subproblems strategy using stochastic discrete velocities. This strategy transforms the originally computationally expensive problem into a series of independently and efficiently solvable subproblems. To be concrete, the proposed method replaces the conventional large deterministic velocity set with multiple small random velocity sets. Each random set defines a subproblem, which is solved by a deterministic multiscale numerical scheme that computes macroscopic moments via Monte Carlo integration. The final flow field is obtained by arithmetic averaging over all sub-problems. In this work, we employ the discrete unified gas kinetic scheme (DUGKS) for spatial discretization and term the resulting method SDV-DUGKS. To validate the proposed method, several numerical test cases are conducted, including (a) the one-dimensional shock structure, (b) the two-dimensional cavity flow, and (c) supersonic flow around a square cylinder. The results of the one-dimensional shock structure confirm the feasibility of the proposed method. The two-dimensional cases demonstrate that, compared to its deterministic counterpart, the proposed method saves more than 80% of memory usage while maintaining comparable accuracy. These results indicate that the proposed method markedly reduces memory demand for multiscale flow simulations and exhibits strong potential to alleviate the curse of dimensionality that currently hinders deterministic multiscale numerical schemes from being applied to engineering problems.