PPM+W, an improved version of the Piecewise Parabolic Method advection scheme: Description, evaluation and implementation in the CHIMERE model.

We have developed a new advection scheme for Eulerian models by combining the Piecewise Parabolic Method “PPM” (Colella, 1984) and the Walcek scheme (Walcek, 2000). The concept is to keep the excellent accuracy of the PPM scheme in the areas of smooth gradient, and to use the flux modifications of Walcek to improve over the PPM scheme in the vicinity of extrema. This combination forms a new advection scheme that we call PPM+W (“Piecewise Parabolic Method + Walcek”). We have studied the properties of this new scheme in the idealized framework of ToyCTM v1.0.1, an academic model including advection and stiff chemistry mimicking tropospheric photochemistry, including inert species and reactive species such as nitrogen oxide, ozones and radicals. This study shows that PPM+W reduces advection error by 10-30% compared to both the Walcek scheme and the PPM scheme. The efficiency of PPM+W is also good: the computation time for PPM+W is not higher than in the classical PPM scheme. Following these results, PPM+W has been implemented in the CHIMERE chemistry-transport model (CHIMERE v2023r1), and could also be useful for other models including atmospheric and oceanic models.

Apart from the design of this new advection scheme, we present a novel method to evaluate advection schemes in contexts where mixing is weak / negligible, and show that this novel method gives results consistent with more classical methods, without requiring the knowledge of the exact solution.

References:
 Colella, P. and Woodward, P. R.: The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54, 174-201, https://doi.org/10.1016/0021-9991(84)90143-8, 1984
 
 Mailler, S., Pennel, R., Menut, L., and Cholakian, A.: An improved  version of the piecewise parabolic method advection scheme: description  and performance assessment in a bidimensional test case with stiff  chemistry in toyCTM v1.0.1, Geosci. Model Dev., 16, 7509–7526,  https://doi.org/10.5194/gmd-16-7509-2023, 2023

Walcek, C., Minor flux adjustment near mixing ratio extremes for simplified yet highly accurate monotonic calculation of tracer advection, J. Geophys. Res., 105, 9335–9348, https://doi.org/10.1029/1999JD901142, 2000