Spatial Moments of Continuous Transport Problems Computed on Grids: Time-Dependent Problems

We present an extension of our recent examination of the method of moments when moments are computed on a spatial grid instead of through integration over the entire domain to include time dependence. For the problem we consider, we show that (i) the flux-weighted average of x remains equal to its initial value but (ii) the flux-weighted average of (x − xa )2 differs from its initial value by an additional error term when moments are determined in this manner, where x is the spatial variable and xa is an arbitrary point. We also present numerical examples that confirm the accuracy of our results.