Determining the effective resolution of advection schemes. Part II: Numerical testing

Numerical models of fluid flows calculate the resolved flow at a given grid resolution. The smallest wave resolved by the numerical scheme is deemed the effective resolution. Advection schemes are an important part of the numerical models used for computational fluid dynamics. For example, in atmospheric dynamical cores they control the transport of tracers. For linear schemes solving the advection equation, the effective resolution can be calculated analytically using dispersion analysis. Here, a numerical test is developed that can calculate the effective resolution of any scheme (linear or non-linear) for the advection equation. The tests are focused on the use of non-linear limiters for advection schemes. It is found that the effective resolution of such non-linear schemes is very dependent on the number of time steps. Initially, schemes with limiters introduce large errors. Therefore, their effective resolution is poor over a small number of time steps. As the number of time steps increases the error of non-linear schemes grows at a smaller rate than that of the linear schemes which improves their effective resolution considerably. The tests highlight that a scheme that produces large errors over one time step might not produce a large accumulated error over a number of time steps. The results show that, in terms of effective-resolution, there is little benefit in using higher than third-order numerical accuracy with traditional limiters. The use of weighted essentially non-oscillatory (WENO) schemes, or relaxed and quasi-monotonic limiters, which allow smooth extrema, can eliminate this reduction in effective resolution and enable higher than third-order accuracy.