TY - JOUR

T1 - Godunov scheme for numerical solution of incompressible Navier-Stokes equations

AU - Kocharina, A. R.

AU - Chirkov, D. V.

N1 - This work was supported by the Russian Science Foundation under grant No. 25-21-00195, https://rscf.ru/en/project/25-21-00195/ This document is a partial result of the research project funded by the Russian Science Foundation under grant No. 25-21-00195, https://rscf.ru/en/project/25-21-00195/

PY - 2026/1/15

Y1 - 2026/1/15

N2 - The incompressible Navier-Stokes equations are solved using the finite-volume artificial compressibility method. A Godunov-type scheme with an exact Riemann solver is developed for the evaluation of inviscid fluxes across cell faces. To this end, the exact solution of the one-dimensional Riemann problem for the artificial compressibility equations is obtained using the method of (u,p)-diagrams. The uniqueness of the solution is rigorously proven. The method is then extended to the multidimensional case. Two approaches for evaluation of the tangential velocity component are examined and discussed. A high-order variant of the Godunov scheme based on third-order MUSCL interpolation is proposed. At that, non-uniformity of the grid is taken into account. An implicit formulation of the scheme is developed, and the linearization process is described in detail. The proposed scheme is compared with the well-established Roe scheme through a series of steady-state two-dimensional benchmark problems, including inviscid and viscous flows around a circular cylinder and the 2D lid-driven cavity flow. The performance of the schemes on non-orthogonal grids is also investigated. Finally, both Roe and Godunov schemes are applied to the simulation of a three-dimensional turbulent flow in a hydraulic turbine flow passage. The results show that while both schemes exhibit comparable accuracy and convergence, the Godunov scheme offers advantages for inviscid simulations on highly non-orthogonal grids.

AB - The incompressible Navier-Stokes equations are solved using the finite-volume artificial compressibility method. A Godunov-type scheme with an exact Riemann solver is developed for the evaluation of inviscid fluxes across cell faces. To this end, the exact solution of the one-dimensional Riemann problem for the artificial compressibility equations is obtained using the method of (u,p)-diagrams. The uniqueness of the solution is rigorously proven. The method is then extended to the multidimensional case. Two approaches for evaluation of the tangential velocity component are examined and discussed. A high-order variant of the Godunov scheme based on third-order MUSCL interpolation is proposed. At that, non-uniformity of the grid is taken into account. An implicit formulation of the scheme is developed, and the linearization process is described in detail. The proposed scheme is compared with the well-established Roe scheme through a series of steady-state two-dimensional benchmark problems, including inviscid and viscous flows around a circular cylinder and the 2D lid-driven cavity flow. The performance of the schemes on non-orthogonal grids is also investigated. Finally, both Roe and Godunov schemes are applied to the simulation of a three-dimensional turbulent flow in a hydraulic turbine flow passage. The results show that while both schemes exhibit comparable accuracy and convergence, the Godunov scheme offers advantages for inviscid simulations on highly non-orthogonal grids.

KW - Artificial compressibility method

KW - Exact solution

KW - Godunov scheme

KW - MUSCL-scheme

KW - Navier-Stokes equations of incompressible fluid

KW - Riemann problem

UR - https://www.scopus.com/pages/publications/105019644000

UR - https://www.mendeley.com/catalogue/f5ce03c2-d9b7-3384-bbec-b6f02ff79399/

U2 - 10.1016/j.compfluid.2025.106881

DO - 10.1016/j.compfluid.2025.106881

M3 - Article

VL - 304

JO - Computers and Fluids

JF - Computers and Fluids

SN - 0045-7930

M1 - 106881

ER -