Periodic boundary conditions on staggered grids: Uniqueness of variables at cell edges/faces

TL;DR

This paper investigates the impact of periodic boundary conditions on staggered grids, revealing that variable uniqueness at cell edges depends on the oddness of the number of points in the periodic direction, due to matrix rank properties.

Contribution

It demonstrates that variable uniqueness on staggered grids with periodic boundaries is guaranteed only when the number of points in the periodic direction is odd, linking this to matrix rank analysis.

Findings

Uniqueness depends on the oddness of point count in the periodic direction.

Matrix rank analysis explains the conditions for variable uniqueness.

Odd number of points ensures full rank and uniqueness.

Abstract

Periodic boundary conditions when applied to staggered grids, which define variables on both cell edges and cell centers, can be shown to have a problem with uniqueness of variables at cell edges depending on the number of points in the direction of periodicity. In the context of the grid defined in this work, it can be shown that uniqueness is guaranteed if and only if the number of points in the periodic direction are odd. This stems from the rank of the matrix with dimensions (N-2) x (N-2) that transforms the values at cell centers to values at edges. This matrix is full rank if and only if N is odd. Here, N is the number of points describing the cell edges.