A comparative study of high-order, quasi-conservative, semi-Lagrangian, and Eulerian advection schemes

Quasi-conservative high-order semi-Lagrangian advection schemes are compared with several positive-definite Eulerian schemes in flux form, including Bott's scheme. In this study, the conventional equipartition method is modified as a posterior iterative mass correction algorithm to restore conservation for semi-Lagrangian transport. The performance comparisons between semi-Lagrangian and Eulerian schemes are evidence that the fifth-order (seventh-order) Bott's area-preserving algorithm without flux limitation is practically equivalent to the quintic (seventh-order) semi-Lagrangian scheme in the rotational flow, with the maximum directional Courant number smaller than 0.5. For positive-definite advection, Bott's algorithm with the flux limitation obtains slightly better (worse) amplitude preservation in the rotational flow tests compared to semi-Lagrangian schemes of same order with (without) the mass correction algorithm. In the nonlinear deformational flow where the maximum directional Courant number is greater than 0.5, the former is slightly unstable and only the short-term results are acceptable, but the latter at the long-term remains stable, conservative, and reasonable. Monotonic tests of semi-Lagrangian schemes were also conducted. It was found that quasi-monotone schemes based on a posterior monotonicity constraint are not influenced by the mass correction algorithm. Both monotonicity and mass conservation can be achieved simultaneously for semi-Lagrangian transport by the post adjustments. However, the monotonicity constraint itself cannot fully suppress the numerical dispersion in the strong nonlinear deformational flow where the mass correction procedure appears to be significantly important for strict mass conservation and a reduction in phase errors.