5.5 Non-Linear Equations and their Solution
119
This kind of under-relaxation was proposed by Patankar (1980). It has a
positive effect on many iterative solution methods since the diagonal dominance of the matrix A is increased (the element A; is larger than Ap, while
Al remains the same). It is more efficient than explicit application of the
expression (5.68).
Optimum under-relaxation factors are problem dependent. A good strategy is to use a small under-relaxation factor in the early iterations and increase it towards unity as convergence is approached. Some guidance for
selecting the under-relaxation factors for solving the Navier-Stokes equations
will be given in Chaps. 7 and 8. Under-relaxation may be applied not only to
the dependent variables but also to individual terms in the equations. It is
often necessary to do so when the fluid properties (viscosity, density, Prandtl
number etc.) depend on the solution and need be updated.
We mentioned above that iterative solution methods can often be regarded
as solving an unsteady problem until a steady state is reached. Control of
the time step is then important in controlling the evolution of the solution.
In the next chapter we shall show that time step may be interpreted as an
under-relaxation factor. The under-relaxation scheme described above may
be interpreted as using different time steps at different nodes.
5.5 Non-Linear Equations and their Solution
As mentioned above, there are two types of techniques for solving non-linear
equations: Newton-like and global. The former are much faster when a good
estimate of the solution is available but the latter are guaranteed not to
diverge; there is a trade-off between speed and security. Combinations of the
two methods are often used. There is a vast literature devoted to methods
for solving non-linear equations but the state-of-the-art is still not completely
satisfactory. We cannot cover even a substantial fraction of the methods here
and give a short overview of some methods.
5.5.1 Newton-like Techniques
The master method for solving non-linear equations is Newton's method.
Suppose that one needs to find the root of a single algebraic equation f (x) =
0. Newton's method linearizes the function about an estimated value of x
using the first two terms of the Taylor series:
f (x) m f (xo) + f 1 ( x o ) b - 20) .
(5.71)
Setting the linearized function equal to zero provides a new estimate of the
root:
f (so)
x1 = xo - - or, in general, xk = xk-1 - f (xk-1)
(5.72)
fl(xo)
fl(xk-1
119
This kind of under-relaxation was proposed by Patankar (1980). It has a
positive effect on many iterative solution methods since the diagonal dominance of the matrix A is increased (the element A; is larger than Ap, while
Al remains the same). It is more efficient than explicit application of the
expression (5.68).
Optimum under-relaxation factors are problem dependent. A good strategy is to use a small under-relaxation factor in the early iterations and increase it towards unity as convergence is approached. Some guidance for
selecting the under-relaxation factors for solving the Navier-Stokes equations
will be given in Chaps. 7 and 8. Under-relaxation may be applied not only to
the dependent variables but also to individual terms in the equations. It is
often necessary to do so when the fluid properties (viscosity, density, Prandtl
number etc.) depend on the solution and need be updated.
We mentioned above that iterative solution methods can often be regarded
as solving an unsteady problem until a steady state is reached. Control of
the time step is then important in controlling the evolution of the solution.
In the next chapter we shall show that time step may be interpreted as an
under-relaxation factor. The under-relaxation scheme described above may
be interpreted as using different time steps at different nodes.
5.5 Non-Linear Equations and their Solution
As mentioned above, there are two types of techniques for solving non-linear
equations: Newton-like and global. The former are much faster when a good
estimate of the solution is available but the latter are guaranteed not to
diverge; there is a trade-off between speed and security. Combinations of the
two methods are often used. There is a vast literature devoted to methods
for solving non-linear equations but the state-of-the-art is still not completely
satisfactory. We cannot cover even a substantial fraction of the methods here
and give a short overview of some methods.
5.5.1 Newton-like Techniques
The master method for solving non-linear equations is Newton's method.
Suppose that one needs to find the root of a single algebraic equation f (x) =
0. Newton's method linearizes the function about an estimated value of x
using the first two terms of the Taylor series:
f (x) m f (xo) + f 1 ( x o ) b - 20) .
(5.71)
Setting the linearized function equal to zero provides a new estimate of the
root:
f (so)
x1 = xo - - or, in general, xk = xk-1 - f (xk-1)
(5.72)
fl(xo)
fl(xk-1