Reconstruction Gradient Limiting

For each cell-0, a limited reconstruction gradient is needed, such that when used in the above reconstruction formula, the reconstructed face value will not exceed the maximum and minimum of the neighboring cell centroid values, including the value in cell-0. Let us define a scale factor that expresses the ratio of the limited and unlimited values, that is:

(364)

For each cell-0, the quantities are defined:

(365)

(366)

where represents the cell value in each neighbor that has a common face with cell-0.

These quantities can also be defined:

(367)

(368)

For each face f of cell-0, is defined:

(369)

Now defining:

(370)

The limiter of Venkatakrishnan [54] gives for the face:

(371)

The cell value of is given by:

(372)