Efficient implicit time-marching schemes with high-order stencils for compressible flow

Abstract

Time-marching techniques are a cornerstone of Computational Aeroacoustics, but existing approaches face significant challenges. Explicit schemes, though straightforward, often require excessively small time steps to maintain numerical stability, while traditional implicit methods achieve larger time steps at the cost of high computational expense per step. This research presents a novel implicit framework that combines high-order differencing stencils for enhanced physical accuracy with low-order preconditioning to ensure numerical stability and efficiency. A comprehensive stability analysis is conducted for preconditioned implicit formulations in both inviscid and viscous flow regimes. The framework is validated across a range of benchmark problems, including one-dimensional, two-dimensional, and three-dimensional inviscid compressible flows, as well as a one-dimensional viscous problem. Results demonstrate the scheme’s ability to achieve robust stability, high accuracy, and reduced computational overhead compared to traditional implicit methods, making it a promising approach for high-fidelity aeroacoustic simulations.