On the computational efficiency of certain upwinding schemes

The computational efficiency of certain upwinding schemes for the discretization of the one-dimensional advection dominated transport equation are examined. In particular, the first order upwind, the second order upwind, the weighted scheme, and the QUICK scheme are considered. The total computational cost for a prescribed accuracy of these schemes was compared using a general purpose algorithm for the integration of time dependent partial differential equations using the method of lines. The method of lines approach enables us to utilize the well developed methodology for the assessment of computational efficiency of ODE integrators. Moreover, in the time integration, the resulting system of ODEs is integrated by an efficient variable order/stepsize numerical integration scheme. The results indicate that for small cell Peclet number, central differences are superior. For moderate values of the cell Peclet number the QUICK scheme is the most attractive. Under high advective flow conditions, however, the second order upwinding is the most reliable.