Search for: All records

Abstract With nested grids or related approaches, it is known that numerical artifacts can be generated at the interface of mesh refinement. Most of the existing methods of minimizing these artifacts are either problem‐dependent or numerical methods‐dependent. In this paper, we propose a universal predictor‐corrector approach to minimize these artifacts. By its construction, the approach can be applied to a wide class of models and numerical methods without modifying the existing methods but instead incorporating an additional step. The idea is to use an additional grid setup with a refinement interface at a different location, and then to correct the predicted state near the refinement interface by using information from the other grid setup. We give some analysis for our method in the setting of a one‐dimensional advection equation, showing that the key to the success of the method depends on an optimized way of choosing the weight functions, which determine the strength of the corrector at a certain location. Furthermore, the method is also tested in more general settings by numerical experiments, including shallow water equations, multi‐dimensional problems, and a variety of underlying numerical methods including finite difference/finite volume and spectral element. Numerical tests suggest the effectiveness of the method on reducing numerical artifacts due to mesh refinement. 

Non‐smoothness arises at cloud edge because, in moist thermodynamics, the thermodynamic properties of the atmosphere are different inside a cloud versus in clear air. In particular, inside a cloud, the vapor pressure of water is constrained by the saturation vapor pressure, which acts as a threshold. Due to this threshold, while the water vapor mixing ratio may vary continuously across cloud edge, its derivatives are not necessarily continuous at cloud edge. Similarly, non‐smoothness also arises for buoyancy and other variables. Consequently, this non‐smoothness in buoyancy and other variables can cause a degraded accuracy in computational simulations. Here we consider special treatment of numerical methods for the interface that arises from phase changes and cloud edges, in order to enhance the accuracy and potentially achieve second‐order accuracy. Numerical solutions are computed for the moist non‐precipitating Boussinesq equations as an idealized cloud‐resolving model with phase changes of water, that is, with cloud formation. Convergence tests, both spatial and temporal, are conducted to measure the numerical error as the grid spacing and time step are refined. While approximately second‐order accuracy is seen in root‐mean‐square (L2) error, the accuracy is degraded in the maximum (Linfinity) error. Discussion is also included on theoretical issues and potential implications for numerical simulations.