5.5 Non-Linear Equations and their Solution
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high that, even though the method does converge in just a few iterations, the
overall cost is greater than that of other iterative methods.
For generic systems of non-linear equations, secant methods are much
more effective. For a single equation, the secant method approximates the
derivative of the function by the secant drawn between two points on the
curve. This method converges more slowly than Newton's method, but as it
does not require evaluation of the derivative, it may find the solution at lower
overall cost and can be applied to problems in which direct evaluation of the
derivative is not possible. There are a number of generalizations of the secant
method to systems, most of which are quite effective but, as they have not
been applied in CFD, we shall not review them here.
5.5.2 Other Techniques
The usual approach t o the solution of coupled non-linear equations is the
sequential decoupled method described in the previous section. The nonlinear terms (convective flux, source term) are usually linearized using Picard
iteration approach. For the convective terms, this means that the mass flux
is treated as known, so the non-linear convective term in the equation for the
u i momentum component is approximated by:
where index o denotes that the values are taken from the result of the previous
outer iteration. Similarly, the source term is decomposed into two parts:
The portion 60 is absorbed into the right hand side of the algebraic equation,
while bl contributes to the coefficient matrix A. A similar approach can be
used for the non-linear terms that involve more than one variable.
This kind of linearization requires many more iterations than a coupled
technique using Newton-like linearization. However, the number of outer iterations can be reduced using multigrid techniques, which makes this approach
attractive.
Newton's method is sometimes used to linearize the non-linear terms; for
example, the convective term in the equation for the u i momentum component can be expressed as:
Non-linear source terms can be treated in the same way. This leads to a coupled linear system of equations which is difficult to solve, and the convergence
is not quadratic unless the full Newton technique is used. However, special
coupled iterative techniques which benefit from this linearization technique
can be developed, as shown by Galpin and Raithby (1986).
121
high that, even though the method does converge in just a few iterations, the
overall cost is greater than that of other iterative methods.
For generic systems of non-linear equations, secant methods are much
more effective. For a single equation, the secant method approximates the
derivative of the function by the secant drawn between two points on the
curve. This method converges more slowly than Newton's method, but as it
does not require evaluation of the derivative, it may find the solution at lower
overall cost and can be applied to problems in which direct evaluation of the
derivative is not possible. There are a number of generalizations of the secant
method to systems, most of which are quite effective but, as they have not
been applied in CFD, we shall not review them here.
5.5.2 Other Techniques
The usual approach t o the solution of coupled non-linear equations is the
sequential decoupled method described in the previous section. The nonlinear terms (convective flux, source term) are usually linearized using Picard
iteration approach. For the convective terms, this means that the mass flux
is treated as known, so the non-linear convective term in the equation for the
u i momentum component is approximated by:
where index o denotes that the values are taken from the result of the previous
outer iteration. Similarly, the source term is decomposed into two parts:
The portion 60 is absorbed into the right hand side of the algebraic equation,
while bl contributes to the coefficient matrix A. A similar approach can be
used for the non-linear terms that involve more than one variable.
This kind of linearization requires many more iterations than a coupled
technique using Newton-like linearization. However, the number of outer iterations can be reduced using multigrid techniques, which makes this approach
attractive.
Newton's method is sometimes used to linearize the non-linear terms; for
example, the convective term in the equation for the u i momentum component can be expressed as:
Non-linear source terms can be treated in the same way. This leads to a coupled linear system of equations which is difficult to solve, and the convergence
is not quadratic unless the full Newton technique is used. However, special
coupled iterative techniques which benefit from this linearization technique
can be developed, as shown by Galpin and Raithby (1986).