## 5.14Descent methods

The Gauss-Seidel method, introduced in Sec. 5.2 , provides convergent solutions for many problems in CFD. It is most eﬀective when a modest reduction in residual is required, e.g. as part of a steady-state solution described in Sec. 5.12 .

When the Gauss-Seidel method requires a lot of sweeps (e.g. over 10) to converge to a suitable tolerance, alternative methods may be more eﬃcient. Descent methods provide alternative matrix solvers that are often used in CFD.

Descent methods represent the equations which are being solved as a minimisation problem. It is demonstrated below using a matrix equation of the form with 2 values

 (5.27)
The minimisation presents the equation in quadratic form; in matrix notation, it is a scalar function of the form
 (5.28)
The quadratic form of Eq. (5.27 ), is:
 (5.29)
 (5.30)
Critically, , i.e. the negative of the residual vector Eq. (5.10 ), as veriﬁed by the model example.

Equating the gradient to zero, , corresponds to a minimum in the quadratic function. At the same time, it is the solution to . The method is therefore concerned with ﬁnding the minimum of the quadratic form eﬃciently.

For this method to work, the quadratic form must have a minimum, which requires that is symmetric and positive-deﬁnite. A positive-deﬁnite matrix is hard to visualise, but for a 2-value function it ensures the quadratic function is a paraboloid.

Diagonal dominance is the convergence condition for the Gauss-Seidel method, discussed in Sec. 5.3 . Importantly, a symmetric matrix that is diagonally dominant, and has positive diagonal coeﬃcients, is also positive-deﬁnite.

### Matrix operations

The ‘’ operation between two single-column matrices, e.g. , in Eq. (5.28 ) is represented in other texts using matrix notation by a transpose . For the example in Eq. (5.27 ), it is:

 (5.31)
Notes on CFD: General Principles - 5.14 Descent methods