Notes on Computational Fluid Dynamics: General Principles

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5.6 Under-relaxation

Sec. 5.5 concludes that the matrix of a typical transport equation is not guaranteed to be diagonally dominant. Some action may therefore be required to ensure a convergent solution.

Under-relaxation is a general method used to improve solution convergence by limiting the amount a variable changes during a solution step.

$PICT\relax \special {t4ht=$

During a solution step, assume a single value of a ﬁeld $\text{[math]}$ in one cell changes from its current value $\text{[math]}$ to the new value $\text{[math]}$ . Under-relaxation would limit the change $\text{[math]}$ by a fraction $\text{[math]}$ , $\text{[math]}$ , so that the value taken from that solution step is

= c + c n = (1 ) + : \relax \special {t4ht=

(5.13)

In some situations, Eq. (5.13 ) is applied after a solution step. This simple approach is known as ﬁeld under-relaxation, which has one notable disadvantage that it requires additional storage of the intermediate ﬁeld $\text{[math]}$ in computer memory.

When a solution step involves solving a matrix equation, the new values $\text{[math]}$ come from an iterative method like Gauss-Seidel. Combining the under-relaxation of Eq. (5.13 ) with the Gauss-Seidel calculation of Eq. (5.4 ) gives:

0 1 c ---BB XN CC i = (1 ) i + ai;i@bi ai;j jA : ji=⇔1j \relax \special {t4ht=

(5.14)

Rearranging Eq. (5.14 ) gives the following relation:

XN 1-ai;i i + ai;j j = bi + 1- 1 ai;i ci: j=1 i⇔j \relax \special {t4ht=

(5.15)

Equation 5.15 is simply the matrix equation $\text{[math]}$ modiﬁed by:

increasing the diagonal coeﬃcients $\text{[math]}$ by division by $\text{[math]}$ ;
multiplying the diﬀerence between the new and original $\text{[math]}$ coeﬃcients by the current $\text{[math]}$ and adding it to the source $\text{[math]}$ .

Modifying the matrix equation this way, known as equation under-relaxation, provides an alternative to Eq. (5.13 ) for under-relaxing a solution of $\text{[math]}$ , without the temporary storage of $\text{[math]}$ .

Ensuring diagonal dominance

The modiﬁcation to $\text{[math]}$ expressed by Eq. (5.15 ) inspires a strategy to ensure diagonal dominance of the matrix as follows.

Each diagonal coeﬃcient which does not satisfy Eq. (5.9 ) is increased until it is diagonally equal. The change to the coeﬃcient is multiplied by the current $\text{[math]}$ and added to $\text{[math]}$ .

This approach to ensure diagonal dominance is eﬀective since it only modiﬁes matrix coeﬃcients where necessary. Otherwise, if the discretisation schemes, and $\text{[math]}$ and $\text{[math]}$ are favourable, then no changes to the matrix are necessary.

Notes on CFD: General Principles - 5.6 Under-relaxation