3.1.4 Other Discretisation Methods 其他离散方法[cfd-3-1-4]

There are few other numerical schemes which are only seldom used in practice, but which are, in certain situations, superior to the methods discussed above. Two particular approaches should be mentioned here briefly.

The first such example is the Spectral Element Method [50]-[53]. The spectral element method combines the geometrical flexibility of the finite element technique with the high-order spatial accuracy (e.g., 10th-order) and the rapid convergence rate of the spectral schemes [54]. The method is based on a high-order polynomial representation of the solution (usually Lagrangian interpolants), combined with a standard Galerkin finite element method, or the method of weighted residuals. The spectral element method is appropriate either for a particular problem in which high-order regularity is guaranteed, or for which high-order regularity is not the exception, like in incompressible fluid mechanics. It is especially suitable for vortical flows. The advantage of the spectral element method is primarily its non-diffusive, non-dispersive approximation of the convection operator, and its good approximation of convection-diffusion boundary layers. The method can treat geometrically and physically complex problems, supposed the condition of high-order regularity is fulfilled. Apart from the rather narrow range of applications, the principal disadvantage of the spectral element method is its very high numerical effort as compared for example to the finite volume method.

Another discretisation scheme, which gained recently some interest, is the so-called Gridless Method [55]-[57]. This method employs only clouds of points for the spatial discretisation. It does not require that the points are connected to form a grid as in conventional structured or unstructured grid schemes. The gridless method is based on the differential form of the governing equations, written in the Cartesian coordinate system. Gradients of the flow variables are determined by a least-squares reconstruction, using a specified number of neighbours surrounding the particular point. The gridless method is neither a finite difference nor a finite volume or a finite element approach since coordinate transformations, face areas or volumes do not have to be computed. It can be viewed as a mix between the finite difference and the finite element method. The principal advantages of the gridless method are its flexibility in solving flows about complex configurations (similar to unstructured methods), and the possibility to locate or cluster the points (or the clouds of points) where it is appropriate. For example, it would be easily possible to select only the neighbours in the characteristic directions when computing gradients. However, there is one unresolved problem. Although the gridless method solves the conservation law form of the Euler or the Navier-Stokes equations, it is not clear whether conservation of mass, momentum and energy is really ensured.

Whichever spatial discretisation scheme we might select, it is important to ensure that the scheme is consistent, i.e., that it converges to the solution of the discretised equations, when the grid is sufficiently refined. It is therefore very important to check how much the solution changes, if the grid is refined (e.g., if we would double the number of grid points). If the solution improves only marginally, we speak of grid converged solution. Another rather self-evident requirement is that the discretisation scheme possesses the order of accuracy, which is appropriate for the flow problem being solved. This rule is sometimes given up in favour of faster convergence, particularly in industrial environment (bad solution is better than no solution). This is of course a very dangerous practice. We shall return to the question of accuracy, stability and consistency later in Chapter 10.