### 11.2.2. Linear Equation Solution

ANSYS CFX uses a Multigrid (MG) accelerated Incomplete Lower Upper (ILU) factorization technique for solving the discrete system of linearized equations. It is an iterative solver whereby the exact solution of the equations is approached during the course of several iterations.

The linearized system of discrete equations described above can be written in the general matrix form:

| (11–49)

where is the coefficient matrix, the solution vector and the right hand side.

The above equation can be solved iteratively by starting with an approximate solution, , that is to be improved by a correction, , to yield a better solution, , that is,

| (11–50)

where is a solution of:

| (11–51)

with , the residual, obtained from:

| (11–52)

Repeated application of this algorithm will yield a solution of the desired accuracy.

By themselves, iterative solvers such as ILU tend to rapidly decrease in performance as the number of computational mesh elements increases. Performance also tends to rapidly decrease if there are large element aspect ratios present.

#### 11.2.2.1. Algebraic Multigrid

The convergence behavior of many matrix inversion techniques can be greatly enhanced by the use of a technique called ‘multigrid’. The multigrid process involves carrying out early iterations on a fine mesh and later iterations on progressively coarser virtual ones. The results are then transferred back from the coarsest mesh to the original fine mesh.

From a numerical standpoint, the multigrid approach offers a significant advantage. For a given mesh size, iterative solvers are efficient only at reducing errors that have a wavelength of the order of the mesh spacing. So, while shorter wavelength errors disappear quite quickly, errors with longer wavelengths, of the order of the domain size, can take an extremely long time to disappear. The Multigrid Method bypasses this problem by using a series of coarse meshes such that longer wavelength errors appear as shorter wavelength errors relative to the mesh spacing. To prevent the need to mesh the geometry using a series of different mesh spacings, ANSYS CFX uses Algebraic Multigrid.

Algebraic Multigrid [25] forms a system of discrete equations for a coarse mesh by summing the fine mesh equations. This results in virtual coarsening of the mesh spacing during the course of the iterations, and then re-refining the mesh to obtain an accurate solution. This technique significantly improves the convergence rates. Algebraic Multigrid is less expensive than other multigrid methods because the discretization of the nonlinear equations is performed only once for the finest mesh.

ANSYS CFX uses a particular implementation of Algebraic Multigrid called Additive Correction. This approach is ideally suited to the CFX-Solver implementation because, it takes advantage of the fact that the discrete equations are representative of the balance of conserved quantities over a control volume. The coarse mesh equations can be created by merging the original control volumes to create larger ones as shown below. The diagram shows the merged coarse control volume meshes to be regular, but in general their shape becomes very irregular. The coarse mesh equations thus impose conservation requirements over a larger volume and in so doing reduce the error components at longer wavelengths.

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