3.1.5 Central and Upwind Schemes 中心格式与上风格式[cfd-3-1-5]

So far, we discussed only the basic choices which exist for the spatial discretisation. But within each of the above three main methods - finite difference, finite volume and finite element - various numerical schemes exist to perform the spatial discretisation. In this context, it is convenient to differentiate between the discretisation of the convective and the viscous fluxes (\(\vec{F}_c\) and \(\vec{F}_v\) in Eq. (2.19), respectively). Because of the physical nature of the viscous fluxes, the only reasonable way is to employ central differences (central averaging) for their discretisation. Thus, their discretisation on structured grids is straightforward. On unstructured triangular or tetrahedral grids, the viscous fluxes are best approximated using the Galerkin finite element methodology, even in the case of a finite volume scheme [58]. The situation becomes more complicated for unstructured mixed grids, where a modified averaging of gradients is more appropriate [59]-[63].

However, the real variety is found in the discretisation of the convective fluxes. In order to classify the individual methodologies, we will restrict our attention to schemes developed for the finite volume method, although most of the concepts are also directly applicable to the finite difference or the finite element method.

To the first category we may count schemes, which are based solely on central difference formulae or on central averaging, respectively. These are denoted as central schemes. The principle is to average the conservative variables to the left and to the right in order to evaluate the flux at a side of the control volume. Since the central schemes cannot recognise and suppress an odd-even decoupling of the solution (i.e., the generation of two independent solutions of the discretised equations), the so-called artificial dissipation (because of its similarity to the viscous terms) has to be added for stabilisation. The most widely known implementation is due to Jameson et al. [64]. On structured grids, it is based on a blend of 2nd- and 4th-differences scaled by the maximum eigenvalue of the convective flux Jacobian. A combination of an undivided Laplacian and biharmonic operator is employed on unstructured grids [65]. The scheme can be improved remarkably using different scaling factors for each equation. This approach is known as the matrix dissipation scheme [66]. It should be mentioned that on unstructured, mixed element grids the explicit Runge-Kutta time-stepping scheme can become unstable, when combined with the conventional central scheme [67].

On the other hand, there are more advanced spatial discretisation schemes, which are constructed by considering the physical properties of the Euler equations. Because they distinguish between upstream and downstream influences (wave propagation directions), they are termed upwind schemes. They can be roughly divided into four main groups:

• flux-vector splitting,
• flux-difference splitting,
• total variation diminishing (TVD), and
• fluctuation-splitting schemes.

Each of these is described briefly in the following subsections.

One class of the flux-vector splitting schemes decomposes the vector of the convective fluxes into two parts according to the sign of certain characteristic variables, which are in general similar to but not identical to the eigenvalues of the convective flux Jacobian. The two parts of the flux vector are then discretised by upwind biased differences. The very first flux-vector splitting schemes of this type were developed in the beginning of the 1980's by Steger and Warming [68] and by Van Leer [69], respectively. A second class of flux-vector splitting schemes decompose the flux vector into a convective and a pressure (an acoustic) part. This idea is utilised by schemes like AUSM (Advection Upstream Splitting Method) of Liou et al. [70], [71], or the CUSP scheme (Convective Upwind Split Pressure) of Jameson [72], [73], respectively. Further similar approaches are the Low-Diffusion Flux-Splitting Scheme (LDFSS) introduced by Edwards [74], or the Mach number-based Advection Pressure Splitting (MAPS) scheme of Rossow [75], [76]. The second group of flux-vector splitting schemes gained recently larger popularity particularly because of their improved resolution of shear layers, but only a moderate computational effort. An advantage of the flux-vector splitting schemes is also that they can be quite easily extended to real gas flows, as opposed to flux-difference splitting or TVD schemes. We shall return to real gas simulations further below.

The second group - flux-difference splitting schemes - is based on the solution of the locally one-dimensional Euler equations for discontinuous states at an interface. This corresponds to the Riemann (shock tube) problem. The values on either side of the interface are generally termed as the left and right state. The idea to solve the Riemann problem at the interface between two control volumes was first introduced by Godunov [77] back in 1959. In order to reduce the numerical effort required for an exact solution of the Riemann problem, approximate Riemann solvers were developed, e.g., by Osher et al. [78] and Roe [79]. Roe's solver is often used today because of its excellent resolution of boundary layers and a crisp representation of shocks. It can be easily implemented on structured as well as on unstructured grids [80].

The idea of TVD schemes was first introduced by Harten [81] in 1983. The TVD schemes are based on a concept aimed at preventing the generation of new extrema in the flow solution. The principal conditions for a TVD scheme are that maxima must be non-increasing, minima non-decreasing, and no new local extrema may be created. Such a scheme is called monotonicity preserving. Thus, a discretisation methodology with TVD properties allows it to resolve a shock wave without any spurious oscillations of the solution. The TVD schemes are in general implemented as an average of the convective fluxes combined with an additional dissipation term. The dissipation term can either depend on the sign of the characteristic speeds or not. In the first case, we speak of an upwind TVD scheme [82], in the second case of a symmetric TVD scheme [83]. The experience shows that the upwind TVD scheme should be preferred since it offers a better shock and boundary layer resolution than the symmetric TVD scheme. The disadvantage of the TVD schemes is that they cannot be easily extended to higher than second-order spatial accuracy. This limitation can be overcome using the ENO (Essentially Non-Oscillatory) discretisation schemes [84]-[89].

The last group - the fluctuation-splitting schemes - provides for true multidimensional upwinding. The aim is to resolve accurately also those flow features which are not aligned with the grid. This is a significant advantage over all above upwind schemes, which split the equations according only to the orientation of the grid cells. Within the fluctuation-splitting methodology, the flow variables are associated with the grid nodes. Intermediate residuals are computed as flux balances over the grid cells, which consists of triangles in 2-D and of tetrahedra in 3-D. The cell-based residuals are then distributed in an upwind-biased manner to the nodes. After that, the solution is updated using the nodal values. In the case of systems of equations (Euler or Navier-Stokes), the cell-based residuals have to be decomposed into scalar waves. Since the decomposition is not unique in 2-D and in 3-D, several approaches were developed in the past. The variety reaches from the wave model of Roe [90], [91] over the algebraic scheme of Sidilkover [92] to the most advanced characteristic decomposition method [93]-[96]. Despite the above mentioned advantage over the dimensionally split Riemann, TVD, etc. solvers, the fluctuation-splitting approaches are so far used only in research codes. This can be attributed to the complexity and the high numerical effort, as well as to convergence problems.

You may ask now, what are the benefits and the drawbacks of the individual spatial discretisation methods. Generally speaking, central schemes require lower numerical effort, and hence less CPU time per evaluation, as compared to upwind schemes. On the other hand, upwind schemes are able to capture discontinuities much more accurately than central schemes. Furthermore, because of their lower numerical diffusion, the upwind schemes can resolve boundary layers using less grid points. Particularly, Roe's flux-difference splitting scheme and the CUSP flux-vector splitting scheme allow a very accurate computation of boundary layers. The negative side of the upwind schemes emerges for second- or higher-order spatial accuracy. The problem is that the so-called limiter functions (or simply limiters) have to be employed in order to prevent the generation of spurious oscillations near strong discontinuities. Limiters are known to stall the convergence of an iteration scheme, because of their accidental switching in smooth flow regions. A remedy was suggested by Venkatakrishnan [97]-[99], which works satisfactorily for most practical cases. However, small wiggles in the solution must be taken into account. Another disadvantage of the limiter functions is that they require high computational effort, particularly on unstructured grids.

With respect to real gas simulations, and in particular to chemically reacting flows, several extensions of the upwind discretisation schemes were presented. For the case of fluids in thermodynamic and chemical equilibrium, modifications of the Van Leer flux-vector splitting [69] and of Roe's approximate Riemann solver [79] were described in Refs. [100]-[102]. Formulations of the both upwind methods for the more complex case of flows with non-equilibrium chemistry and thermodynamics were provided in [103]-[108] and in the references cited therein. In particular, the articles [102], [106] and [107], respectively, give a good overview of the methodologies employed for the upwind discretisation schemes. Recently, a summary of the governing equations together with Jacobian and transformation matrices, which may be required by an upwind scheme, was presented in Ref. [109] for the case of chemically reacting flows.