Taylor-Galerkin

Donea, J., 1984.. A Taylor-Galerkin method for convective transport problems. Int. J. Nwn. Methods Eng. 20, 101 119. 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

O. Colin and M. Rudgyard. Development of high-order taylor-galerkin schemes for unsteady calculations. J. Comput. Phys., 162(2) 338-371, 2000. 

The reason is that the finite element method is not well suited to problems like this, with convection but no diffusion. Your job is to use enough artificial diffusion to eliminate the oscillations in the solution but without obscuring the essential details. A variety of specialized methods are available to do that, as described by Finlayson (1992). The specialized methods include Random Choice, Euler-Lagrange, MacCormack, and Taylor-Galerkin. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Unlike the traditional Taylor s series expansion method, the Galerkin approach utilizes basis functions, such as linear piecewise polynomials, to approximate the true solution. For example, the Galerkin approximation to the sample problem Equation 23.1 would require evaluating Equation 23.13 for the specific grid formation and specific choice of basis function ... 

At this end, to demonstrate superiority of our classical algorithms, we show some sample results of our most recent FDF simulation of the Sandia/Sydney swirl burner [35]. This configuration is selected as it is one of the most challenging turbulent flames for prediction. Figure 3 shows the contours of the azimuthal velocity field as predicted by our FDF. The simulated results agree with experimental data better than any other classical methods currently available [36]. But the computational time requirements are excessive. As another example. Fig. 4 shows the contour of filtered temperature field for the symbolic Taylor-Green vortex flow as obtained via FDF coupled with a discontinuous Galerkin flow solver [37]. Quantum computation may potentially provide a much more efficient means for such simulations. 

The governing equations (9), (10) and (11) are spatially diseretized using a Galerkin finite-element approach in conjunction with P2-P1 tetrahedral Taylor-Hood elements, with ten nodes for velocity and temperature, and four nodes for pressure. A finite element solver for three-dimensional non-Newtonian fluid flow and advection-diffiision heat transfer has been developed. The unknown veloeity, pressure and temperature flelds can be expressed in terms of the shape fimctions and, and nodal... 

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