Advective Schemes 



Numerical treatment of advective terms is a subject of long and 

 continuing interest in the field of numerical analysis. Various second-order 

 schemes and the more complex higher-order schemes have been considered for the 

 present modeling effort. Arakawa, in a series of papers, (Arakawa, 1966, 

 1970; Jespersen, 1974), compared various second-order advective schemes which 

 are classified according to their Jacobians. For example, the J-1 scheme 

 refers to the non-conservative form, the J-3 scheme refers to the conservative 

 form, while the J-6 operator is a combination of J-1 and J-3 schemes. The J-6 

 operator outperforms the others due to its ability to conserve both the mean 

 square vorticity and the kinetic energy in long simulations. Other 

 alternatives also exist, including the fourth-order scheme, the spline method, 

 and the fully implicit advective scheme. 



It was of particular interest to see if the formally more accurate but 

 much more complex fourth-order scheme would allow a reduction in the number of 

 grid points and computational cost to achieve the same order of accuracy as a 

 second-order scheme. For a one-dimensional, two-point boundary value problem 

 and a two-dimensional simulation of the evolution of a Lamb vortex where 

 analytical solutions exist in both cases, our second-order method (the 

 implicit version) was compared with several higher-order methods: (1) the 

 compact implicit block tridiagonal scheme by Kreiss (1975), (2) a similar 

 fifth-order scheme by Thiele (1978), and (3) the operator compact implicit 

 scheme by Hirsh (1975). The tests were performed with a very small time step 

 such that the numerical errors were primarily due to the spatial differencing 

 (Hirsh and Williamson, 1979). The overall efficiency of an advective scheme 

 could be defined as the ratio of the inverse relative error to the amount of 

 computational work (CPU time). It was found that the relative error decreases 

 with the number of grid points, with the fourth-order scheme showing a faster 

 rate of decrease. On the other hand, the computational work increases with 

 the number of grid points, with the fourth order scheme showing a faster rate 

 of increase. The net result is that at a relative error of 5% or more, all 

 the schemes require a comparable number of grid points and have comparable 

 overall efficiencies. The higher-order schemes only gave significantly better 

 overall efficiency at a 1% error level or less. Since it is virtually 



49 



