A Taylor weak statement finite element method for computational fluid dynamics

Taylor weak statement (TWS) or Taylor-Galerkin finite element method approxi- mations were formulated for the advection-diffusion equation, the Burgers equation, and the incompressible Navier-Stokes equations. A new TWS formulation was developed for the shallow-water equations. A Fourier analysis was performed to compare phase ac- curacy and stability for linear basis finite element methods; a new Fourier analysis was also performed for quadratic and cubic basis TWS one-dimensional finite element meth- ods. A multiple-step linear basis TWS finite element method was derived and optimized for phase accuracy. A new modified equation analysis was developed in a general linear form, and the analysis was used to optimize two-dimensional TWS methods for phase ac- curacy and for stability. Phase-accurate TWS finite element methods were found effective in maintaining time accuracy in scalar field equations, but of limited value for the transient Navier-Stokes equation systems tested.

Stability-enhanced artificial diffusion TWS methods for the incompressible Navier- Stokes equations were systematically compared for the first time to a wide variety of modified finite element methods, including SUPG, exponential Petrov-Galerkin, SGM, hierarchical basis, least-squares, and monotone methods. The artificial diffusion meth- ods were tested on various two-dimensional advection-diffusion and incompressible laminar Navier-Stokes benchmark problems. In contrast to the phase-accurate methods, the stability-enhanced methods were found to effectively decrease grid requirements for high Reynolds number laminar flow benchmark problems, including the backward-facing step and cylinder-in-crossflow. The best performing methods were the exponential Petrov- Galerkin and streamline-diffusion (TWS or SUPG) methods. As expected, the artificial diffusion methods were found ineffective on a high Rayleigh number natural convection benchmark problems characterized by small magnitude of velocity.