A Galerkin finite element algorithm based on third‐order Runge‐Kutta temporal discretization along the uniform streamline for unsteady incompressible flows

Abstract

Summary In this paper, for two‐dimensional unsteady incompressible flow, the Navier‐Stokes equations without convection term are derived by the coordinate transformation along the streamline characteristic. The third‐order Runge‐Kutta method along the streamline is introduced to discrete the alternative Navier‐Stokes equations in time, and spacial discretization is carried out by the Galerkin method, and then, the third‐order accuracy finite element method is obtained. Meanwhile, the streamline velocity is uniformly approximated by initial velocity in each time step in order to reduce update frequency of total element matrix and improve calculation efficiency. Finally, some classic unsteady flow examples are calculated and analyzed by different calculation methods, which further demonstrate that the present method has more advantages in stability, permissible time step, dissipation, computational cost, and accuracy. The code can be downloaded at https://doi.org/10.13140/RG.2.2.27706.44484.

A stabilized finite element method based on characteristic‐based polynomial pressure projection scheme for incompressible flows

“…The reason is that difficult establishment of an upwind scheme in FEM to deal with the convection caused oscillation. After decades of devotions, many efficient stabilizations have been proposed for FEM to circumvent this kind of oscillation, such as streamline upwind Petrov–Galerkin , 2,3 stabilized‐space‐time method, 4 Galerkin least square , 5 finite increment calculus , 6 Taylor–Galerkin , 7,8 characteristic‐Galerkin (CG), 9,10 flow‐condition‐based interpolation , 11,12 uniform characteristic‐based Runge–Kutta FEM , 13 etc.…”