Square error estimate for adaptive mesh refinement

In a particular implementation(Finite volume advection using upwind) of adaptive mesh refinement the error square estimate for a cell C is given as

$$\sum_{i = x,y,z} vol * \frac{1}{12} * h^{2} * (\frac{ds}{di})^{2}$$

where $vol$ is the volume of the cell, $h$ is the size of the cell and ${ds}/{di}$ is the derivative of the solution $s$ in each of the direction $i$.

What I still fail to understand that where is this error expression coming from and is the error expression specific for the scheme being implemented which is first order accurate in space or, can it be applied for any other scheme too. I tried to find the truncation error in space for upwind scheme but the expression doesn't matches the expression given above. What i also tried, is to use the same expression for error estimate for a second order accurate central differencing scheme for diffusion scheme( nomal 5 point stencil in 2D or 7 point stencil in 3D) and it seems to work for that though I face feeble issues.

Any other better ideas or pointers for error estimation to dynamically refine the mesh is also welcome.

• You may want to read through chapter 19 of LeVeque, R. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002. – Kyle Mandli Jul 20 '16 at 22:51
• @KyleMandli I just went through ch 19 and 20. 20 does mention the donor cell upwinding but nowhere it is mentioned the truncation error. Sorry bu I have tried deducing it and not able to reach this solution. Maybe i am missing something. Thanks – datapanda Jul 20 '16 at 23:43