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I am following the approach of Hundsdorfer from Numerical solution of time-dependent advection-diffusion-reaction equations in which they introduce adaptive upwinding. The method can adapt from pure upwind ($\kappa=1$) to pure central ($\kappa=0$) discretisation for the advection term (N.B. the diffusion term is always based on a central difference), $$ u_j^{\prime} = \frac{1}{2h}a \left( u_{j-1} - u_{j+1} \right) + \left( \frac{d}{h^2} + \kappa\frac{a}{2h} \right) \left( u_{j-1} - 2u_{j} + u_{j+1} \right) $$

$u_j^{\prime}$ is shorthand for the time derivative, $a$ is the velocity ($a>0$), $d$ is the diffusion coefficient and $h$ is the step size between uniform mesh points.

I wish to extend this to the case of a non-uniform mesh, however I run into a problem because I get a factor of $4$ appearing when I do so, which seems inconsistent. For example, modify the above equation to include non-uniform steps,

$$ u_j^{\prime} = \frac{1}{2h_j^{+}}a \left( u_{j-1} - u_{j+1} \right) + \left( \frac{d}{(h_j^{+} + h_j^{-})^2} + \kappa\frac{a}{2h_j^{+}} \right) \left( u_{j-1} - 2u_{j} + u_{j+1} \right) $$

where $h_j^{+}=x_{j+1} - x_j$ and $h_j^{-}=x_{j} - x_{j-1}$. We can check the consistence by putting this equation back on to a uniform mesh. Substituting $h_j^{+}\rightarrow h$ and $h_j^{-}\rightarrow h$,

$$ u_j^{\prime} = \frac{1}{2h}a \left( u_{j-1} - u_{j+1} \right) + \left( \frac{d}{4h^2} + \kappa\frac{a}{2h} \right) \left( u_{j-1} - 2u_{j} + u_{j+1} \right) $$

This is identical appart form the factor of $4$ appearing in the denominator!

I haven't much experience with these adaptive upwinds, can you see an issue with using a modified version of the original equation,

$$ u_j^{\prime} = \frac{1}{2h}a \left( u_{j-1} - u_{j+1} \right) + \left( \frac{4d}{(2h)^2} + \kappa\frac{a}{2h} \right) \left( u_{j-1} - 2u_{j} + u_{j+1} \right) $$

The additional constants cancel on both a uniform and non-uniform grid, giving the correct central difference diffusion term in all cases,

$$ d\frac{\left( u_{j-1} - 2u_{j} + u_{j+1} \right) }{h^2} $$


Update

Using a Taylor series to calculate the second derivative (the diffusion term) with non-uniform step gives,

$$ 2d \frac{h^{+} \left(- u_{j} + u_{{j-1}}\right) + h^{-} \left(- u_{j} + u_{{j+1}}\right)}{h^{+} h^{-} \left(h^{+} + h^{-}\right)} $$

This reduces to the correct expression when the grid is uniform.

However, this makes the equation hard to write in a compact form because the $h^{-}$ and $h^{+}$ terms appear in the numerator. Does anybody you have any comments on whether this can be simplified further, i.e. so the $h$ terms don't appear in the numerator? This will allow the write the equation in the original form.

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You need to rederive the term $$ \frac{d}{(h_j^+ + h_j^-)^2} $$ because this appears to be wrong. It must lead to factor of 4 when switching to a uniform grid because you've taken the sum of two things that are $O(h)$ and squared it.

Perhaps if you spell out the derivation, we can help you find the error.

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  • $\begingroup$ This is simply just a central finite difference applied to a second order derivative but with non-uniform step. I can do this with a Taylor expansion around, $f(x-h^{-})$ and $f(x+h^{+})$. Thanks for the advice, I'll do that. $\endgroup$ – boyfarrell Jun 9 '13 at 3:40
  • $\begingroup$ OK there is an update above. But the derived equation is hard to write in a compact form! Can it be simplified further? I'm not sure. $\endgroup$ – boyfarrell Jun 9 '13 at 5:40
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    $\begingroup$ I believe your only option is to separate it into two terms with different denominators. That will allow you to cancel the $h^-$ in one case and the $h^+$ in the other, but now you'll have two terms where you had one before. $\endgroup$ – Bill Barth Jun 10 '13 at 3:48

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