Development and application of fourth-order-accurate semi-implicit scheme for Navier-Stokes equations

This paper presents a semi-implicit time discretisation scheme with fourth-order accuracy for the compressible Navier-Stokes equations. The scheme integrates an explicit basis framework with implicit components to enhance stability while maintaining the accuracy of the original explicit scheme. Combining Runge-Kutta and linear multi-step methods, the explicit basis is designed to meet underdetermined equations with free parameters optimised for implicit requirements. An implicit term adjustable by parameter γ is added at each stage to modify implicitness. The resulting implicit scheme is A-stable, verified through linear stability analysis. Applied to space-time decoupled compressible N-S equations, the LU-SGS method is utilised for the implicit operator matrix to reduce computational cost and improve stability, in conjunction with dual-time stepping. Classical test cases demonstrate that the scheme effectively captures shock waves, low-speed turbulence, and shock wave/boundary layer interactions, ensuring accuracy and stability with large time steps, offering higher efficiency than traditional approaches.