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I have some PDE system involving unknowns $u$ and $v$ and $w$ which are concentrations of some substances. I want to nondimensionalize the system, so I divide through $u$ by some number $c_u$, and similarly for $v$ and $w$ with $c_v$ and $c_w$.

The numbers $c_u$ etc. is a typical concentration value for the corresponding variable. I have been using initial data for the model system given in a paper to choose what these numbers should be.

But unfortunately the initial data for one of these is given to be zero, i.e. $c_v = 0$.

What is the standard step to do in this case? How to get around it?

25Chars
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1 Answers1

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If $u,v,w$ are concentrations measured in moles per litre you can non-dimensionalize them by dividing them all by $b$ moles per litre so you have non-dimensional concentrations $u^*=\frac{u}{b}, v^*= \frac{v}{b}, w^*=\frac{w}{b}$.

You can then solve the PDE equations in terms of $b$.

If you need a numerical value of $b$ in order to solve the PDE numerically then for simplicity you can set b equal to 1 mole per litre so there is effectively no change to the equations.

Edit: Although $b=1$ is simple, the reason we usually use $c_u$ (the typical concentration you mention) is because it is of a similar magnitude to the concentrations $u$ could be. By taking $b=1$ mole per litre we could end up with very small values for non-dimensional concentrations if $u$ was normally 1 millimole per cubic metre. These very small or very large numbers can cause numerical instability if using numerical methods to solve the PDE.

Hugh
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  • Thanks. But what if two of the concentrations say $u$ and $w$ are very different: like a difference of $10^5$. – 25Chars Jan 28 '16 at 11:30
  • Well you can use different constants $b_1,b_2,b_3$ to normalize them. I should have mentioned that. – Hugh Jan 28 '16 at 14:07