A note on non‐negativity correction for a multimoment finite‐volume transport model with WENO limiter

Abstract

An oscillation‐less multimoment global transport model was proposed by Tang et al. in 2018 by introducing a slope limiter based on the weighted essentially non‐oscillatory (WENO) concept. The spurious oscillations around the discontinuities and strong gradients can be effectively removed and the model preserves the fourth‐order accuracy in spherical geometry for smooth solutions. However, the WENO limiter does not strictly guarantee the non‐negativity of numerical solution and the resulting model will violate the physical principle due to the existence of the nonphysical negative undershoots. As the numerical fluxes, which determine the time tendency of the volume‐integrated average, are evaluated by the point values (PVs) defined along the cell boundaries in the multimoment finite‐volume schemes, the non‐negativity preserving model can be accomplished by modifying those PVs to implement the corrections on the numerical fluxes proposed in the positive definite flux‐corrected transport (FCT) method by Smolarkiewicz in 1989. In this study, a non‐negativity correction algorithm is proposed for a global transport model using the MM‐FVM_WENO scheme and verified by simulating the widely used benchmark tests.

A three‐dimensional positivity‐preserving and conservative multimoment finite‐volume transport model on a cubed‐sphere grid

“…It should be mentioned that the non‐negativity flux correction transport technique in our previous work (Li et al, 2020) is implemented before each Runge–Kutta substep in order to keep the numerical solution positive‐definite in the horizontal, since the MCV3‐BGS scheme does not ensure a strictly positive solution in spite of its nonoscillatory advantages.…”

“…As a whole, the novelties of this study are as follows: (1) development of a conservative, nonoscillatory, and positive‐definite transport model, which is suitable for real weather and climate simulations; (2) extension of the MCV3‐BGS algorithm (Deng et al, 2017) with non‐negativity correction (Li et al . 2020) into the cubed‐sphere grid in the horizontal; (3) combination of the PRM scheme (Xiao and Peng, 2004) with modified positive‐definite correction (Tang et al, 2021) with a fourth‐order non‐negativity MCV3‐BGS algorithm to assure the positivity of the 3D transport model.…”

“…The resulting model can maintain high‐order accuracy in the horizontal direction and permits a large Courant number in the vertical direction, which is very computationally efficient and beneficial for practical application. Furthermore, two non‐negativity corrections in our previous work (Li et al, 2020; Tang et al, 2021) are implemented in the transport model to ensure a positive numerical solution. As a whole, the novelties of this study are as follows: (1) development of a conservative, nonoscillatory, and positive‐definite transport model, which is suitable for real weather and climate simulations; (2) extension of the MCV3‐BGS algorithm (Deng et al, 2017) with non‐negativity correction (Li et al .…”

A nonnegative and shape‐preserving global transport model on cubed sphere using high‐order conservative collocation scheme

“…It should be noted that the specified condition is strictly preserved only if the nondivergent velocity field is completely satisfied in discrete form. The details of numerical operations can be referred to Li et al 27 …”