B

FIGURE 6.8 Structure of the matrix for a five-point computational molecule.

of the elements are zero). The structure of matrix A depends on the ordering of variables in vector <p and the size of the computational molecule. For the case of a five-point computational molecule, the structure of A is shown in Fig. 6.8. For the unstructured grid, the coefficient matrix A remains sparse, however, it no longer has a banded structure. Different types of solvers are, therefore, needed for structured and unstructured grids. In this section, some of the methods used to solve algebraic equations resulting from the structured grid are discussed. Details of solvers suitable for unstructured grid can be found in Saad and Schultz (1986). General information about the solution of algebraic discretized equations may be found in Ferziger and Peric (1995).

Direct methods of solution of linear algebraic equations are essentially matrix inversion algorithms (Gauss elimination, LU decomposition etc. Details of these methods can be found in Press et al., 1992). These methods have large memory requirements and are computationally expensive for a large number of equations. These methods become especially inefficient when solving linearized non-linear equations. Iterative methods are based on repeated application of a relatively simple algorithm (a Jacobi point by point method or line by line methods) leading to eventual convergence. If each iteration is inexpensive and the required number of iterations is small, an iterative method will be more efficient than the direct method. For many CFD problems, this is usually the case. The other advantage of iterative methods is that only non-zero coefficients of the equations need to be stored in core memory. Some of the basic methods which can be used with iterative solvers are the Gauss-Siedel method, successive over-relaxation (SOR) and the tri-diagonal matrix algorithm (TDMA). The TDMA is actually a direct solver for the one-dimensional problem if the value at one node is a function of only its neighboring two nodes. It is, however, widely used in a line-by-line fashion to solve multidimensional CFD problems (where the value at a node is a function of more than two neighboring nodes). Proper choice of sweeping the lines on which TDMA is applied can accelerate the overall convergence rate (for example, marching the lines from upstream to the downstream direction). When higher order discretization schemes are used (QUICK or second-order upwind schemes), the TDMA method can be applied only by incorporating a large number of neighboring contributions in the source term. Understandably, the performance of TDMA deteriorates for such cases, resulting in slower convergence. A more generalized version of TDMA known as the penta-diagonal algorithm (for a set of linear equations containing five non-zero elements per equation) is available (see Fletcher, 1991).

For even larger computational molecules (arising in multidimensional problems with complex body fitted grids), Stone (1968) proposed a strongly implicit procedure (SIP). Schneider and Zedan (1981) developed an improved version of this method called modified SIP or MSIP. Details of these methods may be found in the cited references and in Anderson et al. (1984) and Peric (1987). These methods may also act as a basis for developing additional methods to accelerate convergence, such as conjugate gradient (Golub and van Loan, 1990) or multi-grid methods (Hackbusch, 1985; Peric et al., 1988; Sathyamurthy and Patankar, 1994). Several acceleration methods based on additive correction philosophy have been proposed (for example, Hutchinson and Raithby, 1986; Kelkar and Patankar, 1989). In additive correction philosophy, a correction to the current solution field is sought, so that the corrected solution obeys global conservation within the considered sub-region. Combination of the correction methods and multi-grid methods may also be effectively used to accelerate the convergence. Multi-grid methods are more like useful strategies than specific methods. Within the multi-grid framework, several choices of such parameters as number of grid levels, number of iterations on each grid level, the order in which various levels are visited, interpolation between various levels, may be made to construct different methods. The rate of convergence, of course, will depend on the choice of these parameters. These details will not be discussed here and may be found in the cited references.

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