The accuracy of the node-centered finite volume method in one-dimension is analyzed. Numerical simulations and analysis are performed for both a hyperbolic and a elliptic case, for various types of grids. The results from the simulations agree with the analysis. The boundary conditions are implemented weakly using penaly technique. For the hyperbolic case we see that the type of grid has large impact on the order of accuracy, whereas the choice of penaly parameter only affect the error constant. For the elliptic case the grid has less impact on the order of accuracy. For both the hyperbolic and elliptic problem we show that the error contribution from the primal and dual grid can be treated separately.