1st or 2nd order gradients approach？
 

hi every one
i have written a 3d unstructured(hybrid) NS solver using a cell-centered finite volume method. three methods of green, unweighted-LSQ and weighted LSQ have implemented to calculate the gradients.after a series of tests on a field which i know analytic solutions of gradients, i found

1）Compare the three methods result with analytic one on regular grid, the relatively error were about 1e-13(machine zero is 1e-14 on my system); and on iregualr grid, the relatively error were about 1e-3 to 1e-5. dose this mean my program works right?

2) on regular grid, all the three methods cannot achieve 2nd order accuracy, the green and unweighted-LSQ achieved 1st order and weighted LSQ was more than 1st; and on iregualr grid the green and unweighted-LSQ cannot achieved 1st order and weighted LSQ was 1st order. i am confused because some people said the three method are 1st order and some said they are 2nd order method, and i want to know dose the gradient accuracy less than 2nd order would have any affect on accuracy of spatial discretization, if i use a linear reconstruction of Barth

There is nothing called first order gradient and second order gradient. Gradient is always first order term. Accuracy of gradient calculation would affect the solution.

The gradient accuracy order for an overall second order FV cell-centered code (linear reconstruction) just need to be first. However, it is important that it is first order on any grid. Still, for fully cartesian, uniform, meshes i expect some methods to return a second order accuracy.

How are you handling your boundary conditions? Is it possible that they are not treated consistently for the analytic solution?